Feynman integral relations from parametric annihilators
Thomas Bitoun Christian Bogner Rene Pascal Klausen Erik Panzer
hep-th math-ph math.MP

Kinematics of CMEs and associated shock waves as deduced from LASCO data: comparative analysis
    Zagainova  Yu.S.
    
==============================================================================================


	We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space integration by parts relations, which are well-known in the physics literature.
	Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works.











§ INTRODUCTION






At higher orders in perturbative quantum field theory, the computation of observables via Feynman diagrams involves a rapidly growing number of Feynman integrals.
Fortunately, the number of integrals which need to be computed explicitly can be reduced drastically by use of linear relations 

    ∑_i c_i_i = 0
    	∗

between different Feynman integrals _i, with coefficients c_i that are rational functions of the space-time dimension  and the kinematic invariants characterizing the physical process (masses and momenta of elementary particles).

The most commonly used method to derive such identities is the integration by parts (IBP) method introduced in <cit.>. In this approach, the relations (∗) are obtained as integrals of total derivatives in the momentum-space representation of Feynman integrals. Combining these relations, any integral of interest can be expressed as a linear combination of some finite, preferred set of master integrals.[
	For different applications, various criteria for choosing the master integrals have been suggested. These include uniform transcendentality <cit.>, finiteness <cit.> or finiteness of coefficients <cit.>.
]
Laporta's algorithm <cit.> provides a popular approach to obtain such reductions, and various implementations of it are available <cit.>.
However, the increase of complexity of todays computations has recently motivated considerable theoretical effort to improve our understanding of the IBP approach
and the efficiency of automated reductions <cit.>. This includes a method by Baikov <cit.>, which is based on a parametric representation of Feynman integrals.

It is interesting to ask for the number of master integrals which remain after such reductions. This number provides an estimate for the complexity of the computation and informs the problem of constructing a basis of master integrals by an Ansatz. In the recent literature, algorithms to count master integrals were proposed and implemented in the computer programs  <cit.> and  <cit.>. 

We propose an unambiguous definition for the number of master integrals as the dimension of an appropriate vector space. This definition and our entire discussion are independent of the method of the reduction. The main result of this article shows that this number is a well understood topological invariant: the Euler characteristic of the complement of a hypersurface =0 associated to the Feynman graph. 
Therefore, many powerful tools are available for its computation.

To arrive at our result, we follow Lee and Pomeransky <cit.> and view Feynman integrals as a Mellin transform of ^-/2, where  is a certain polynomial in the Schwinger parameters. Each of these parameters corresponds to a denominator (inverse propagators or irreducible scalar products) of the momentum-space integrand.
The classical IBP relations relate Feynman integrals which differ from each other by integer shifts of the exponents of these denominators.
As Lee <cit.> and Baikov <cit.> pointed out, such shift relations correspond to annihilation operators of the integrands of parametric representations.[
	Tkachov's idea <cit.> to insert Bernstein-Sato operators in the integrand with two Symanzik polynomials was used for numerical computations of one- and two-loop integrals 
<cit.>.
]
In our set-up, these are differential operators P satisfying

    P ^-/2 = 0.

We recall that such parametric annihilators provide all shift relations between Feynman integrals, in particular the ones known from the classical IBP method in momentum space. The obvious question, whether the latter suffice to obtain all shift relations, seems to remain open. As a positive indication in this direction, we show that the momentum space relations contain the inverse dimension shift.

Ideals of parametric annihilators are examples of D-modules.
Loeser and Sabbah studied the algebraic Mellin transform <cit.> of holonomic D-modules and proved a dimension formula in <cit.>, which, applied to our case, identifies the number of master integrals as an Euler characteristic.
The key property here is holonomicity, which was studied in the context of Feynman integrals already in <cit.> and of course is crucial in the proof <cit.> that there are only finitely many master integrals.

It is furthermore worthwhile to notice that algorithms <cit.> have been developed to compute generators for the ideal of all annihilators of ^-/2, see also <cit.>.
Today, efficient implementations of these algorithms via Gröbner bases are available in specialized computer algebra systems such as  <cit.>.
We hope that these improvements may stimulate further progress in the application of D-module theory to Feynman integrals <cit.>.

We begin our article with a review of the momentum space and parametric representations of scalar Feynman integrals and recall how the Mellin transform translates shift relations to differential operators that annihilate the integrand.
In section <ref> we illustrate how the classical IBP identities obtained in momentum space supply special examples of such annihilators. The relations between integrals in different dimensions are addressed in section <ref>, where we relate them to the Bernstein-Sato operators and show that these can be obtained from momentum space IBPs.
Our main result is presented in section <ref>, where we apply the theory of Loeser and Sabbah to count the master integrals in terms of the Euler characteristic.
Practical applications of this formula are presented in the following section <ref>, which includes a comparison to other approaches and results in the literature.
Finally we discuss some open questions and future directions.

In appendix <ref> we give an example to illustrate our definitions in momentum space, present proofs of the parametric representations and demonstrate algebraically that momentum space IBPs are parametric annihilators.
The theory of Loeser and Sabbah is reviewed in appendix <ref>, which includes complete, simplified proofs of those theorems that we invoke in section <ref>.
Finally, appendix <ref> discusses the parametric annihilators of a two-loop example in detail.






  
Acknowledgments
We like to thank Mikhail Kalmykov, Roman Lee, Robert Schabinger, Lorenzo Tancredi and Yang Zhang for interesting discussions and helpful comments on integration by parts for Feynman integrals, Jørgen Rennemo for suggesting literature relevant for section <ref> and Viktor Levandovskyy for communication and explanations around D-modules and in particular their implementation in .

Thomas Bitoun thanks Claude Sabbah for feedback and bringing the work <cit.> to our attention. Thomas acknowledges funding through EPSRC grant EP/L005190/1.

Christian Bogner thanks Deutsche Forschungsgemeinschaft for support under the project BO 4500/1-1.

This research was supported by the http://www.munich-iapp.de/Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence “Origin and Structure of the Universe”. Furthermore we are grateful for support from the http://hu.berlin/kmpbKolleg Mathematik Physik Berlin (KMPB) and hospitality at Humboldt-Universität Berlin.

Images in this paper were created with  <cit.> (based on  <cit.>) and  <cit.>.






§ ANNIHILATORS AND INTEGRAL RELATIONS




In this section we elaborate a method to obtain relations between Feynman integrals from differential operators with respect to the Feynman parameters and show that these relations include the well known IBP relations from momentum space.






 §.§ Feynman integrals and Schwinger parameters




At first we fix conventions and notation for Feynman integrals in momentum space and recall their representations using Schwinger parameters. While the former is the setting for most traditional approaches to study IBP identities, it is the latter (in particular in its form with a single polynomial) which provides the direct link to the theory of D-modules that our subsequent discussion will be based on.

We consider integrals (also called integral families <cit.>) that are defined by an -fold integral ( is the loop number), over so-called loop momenta _1,…,_ in -dimensional Minkowski space, of a product of powers of denominators =(_1,…,_N):

    (ν_1,…,ν_N)
    	= (∏_j=1^L∫[] _j/π^/2)
    	∏_a=1^N_a^-ν_a
    	.

The denominators are (at most) quadratic forms in the  loop momenta and some number  of linearly independent external momenta _1,…,_.
In most applications, the denominators are inverse Feynman propagators associated with the momentum flow through a Feynman graph that arises from imposing momentum conservation at each vertex, see example <ref>. However, we will only restrict ourselves to graphs from section <ref> onwards, and keep our discussion completely general until then.

An integral (<ref>) is a function of the indices ν=(ν_1,…,ν_N) (denominator exponents), the dimension  of spacetime and kinematical invariants (masses and scalar products of external momenta). However, we suppress the dependence on kinematics in the notation and treat kinematical invariants as complex numbers throughout.
The dimension and indices are understood as free variables; that is, we consider Feynman integrals as meromorphic functions of (,ν)∈^1+N in the sense of Speer <cit.>.


	
	To each denominator _a of a list =(_1,…,_N), we associate a variable x_a (1≤ a ≤ N) called Schwinger parameter.
	The linear combination

    ∑_a=1^Nx_a_a
    	=-∑_i,j=1^_ij(_i·_j)
    	+∑_i=1^ 2 (_i·_i)
    	+

	decomposes into quadratic, linear and constant terms in the loop momenta.[
		The scalar products _i·_j = _i^1 _j^1 - ∑_μ=2^_i^μ_j^μ are understood with respect to the Minkowski metric.
	]
	This defines a symmetric × matrix , a vector Q of  linear combinations of external momenta and a scalar .
	We define furthermore the polynomials

    ,
    	  
    
    
    	 ( ^^-1 + )
    	      and  +  
    	.


Schwinger parameters yield useful representations of the Feynman integrals (<ref>):
 
	Let us denote the superficial degree of convergence by
	
    ν - /2  where  ν∑_i=1^Nν_i
    		.

	Then the Feynman integral (<ref>) can be written as

    (ν_1,…,ν_N)
    	   =
    	(∏_i=1^N∫_0^∞x_i^ν_i-1 x_i/Γ(ν_i))
    	e^-//^/2
    	,
    	
    	
    (ν_1,…,ν_N)
    	   =Γ()
    	(
    		∏_i=1^N∫_0^∞x_i^ν_i-1 x_i/Γ(ν_i))
    	δ(1-∑_j=1^Nx_j)/^/2-^  and
    	
    (ν_1,…,ν_N)
    	   = Γ(/2)/Γ(/2-)(∏_i=1^N∫_0^∞x_i^ν_i-1 x_i/Γ(ν_i))
    	^-/2
    	.


The formulas (<ref>) and (<ref>) are known since the sixties and we refer to <cit.> for detailed discussions and for the original references. The trivial consequence (<ref>) was popularized only much more recently by Lee and Pomeransky <cit.> and it is this representation that we will use in the following.
In appendix <ref> we include proofs for these equations and provide further technical details.


	
	Consider the graph in figure <ref> with massless Feynman propagators, _1=-^2 and _2=-(+)^2.



	We find = x_1 + x_2, = -x_2 p and = -x_2 p^2 according to (<ref>).
	Hence the graph polynomials (<ref>) become
	= x_1+x_2 and = (-p^2)x_1 x_2 such that
	
    (ν_1,ν_2)
    		= Γ(/2)/Γ(-ν_1-ν_2)∫_0^∞x_1^ν_1-1 x_1/Γ(ν_1)∫_0^∞x_2^ν_2-1 x_2/Γ(ν_2)( x_1 + x_2 -p^2 x_1 x_2 )^-/2
    		.



	Depending on the values of  and ν, the integrals (<ref>) and (<ref>)–(<ref>) can be divergent. However, there exists a non-empty, open domain in ^1+N∋ (,ν) where all of them are convergent and agree with each other.[
	The only exception are cases of zero-scale subintegrals, like massless tadpoles. Such integrals are zero in dimensional regularization and therefore irrelevant for our considerations.
]
	From there, analytic continuation defines a unique, meromorphic extension of every Feynman integral to the whole parameter space ^1+N. The poles are simple and located on affine hyperplanes defined by linear equations with integer coefficients. For these foundations of analytic regularization[
	The widely used dimensional regularization is the special case when ν∈^N are all integers and only the dimension  remains as a regulator. In this case, the poles are not necessarily simple anymore.
] we refer to <cit.>.

	The uniqueness of the analytic continuation of a Feynman integral (as a function of  and ν) is very important; in particular, it means that an identity between Feynman integrals is already proven once it has been established locally in the non-empty domain of convergence of the involved integral representations. 
	In other words, in any calculation with analytically regularized Feynman integrals, we may simply assume, without loss of generality, that the parameters are such that the integrals converge. The resulting relation then necessarily remains true everywhere by analytic continuation.


	The one-loop propagator in example <ref> can be computed in terms of Γ-functions:
	
    (ν_1,ν_2)
    		= (-p^2)^/2-ν_1-ν_2Γ(/2-ν_1)
    			Γ(/2-ν_2)
    			Γ(ν_1+ν_2-/2)
    		/Γ(ν_1)
    			Γ(ν_2)
    			Γ(-ν_1-ν_2)
    		
    		.

	The integral (<ref>) converges only in a certain domain of (ν,), but there it evaluates to (<ref>), which has a unique meromorphic continuation. Its poles lie on the infinite family of hyperplanes defined by /2-ν_1=k, /2-ν_2=k and ν_1+ν_2-/2=k, indexed by k ∈_≤ 0.








 §.§ Integral relations and the Mellin transform




In this section we summarize how relations between ν-shifted Feynman integrals can be identified with differential operators that annihilate ^-/2. This method was suggested in <cit.> (and in <cit.> for the Baikov representation).

The parametric representation (<ref>) can be interpreted as a multi-dimensional Mellin transform. For our purposes, we slightly deviate from the standard definition, as for example given in <cit.>, and include the factors Γ(ν_i) that occur in (<ref>).
 
    
	Let ν=(ν_1,…,ν_N)∈^N. The twisted (multi-dimensional) Mellin transform of a function f_+^N⟶ is defined as 

    f(ν)
    	(∏_i=1^N∫_0^∞x_i^ν_i-1 x_i/Γ(ν_i))f(x_1,…,x_N),

	whenever this integral exists. As a special case we define
	
    (ν)
    		^-/2(ν)
    		  such that   (ν)
    		= Γ(/2)/Γ(/2-)(ν)
    		.


Recall that, as mentioned in remark <ref>, we do not have to worry about the actual domain of convergence of (<ref>) in the algebraic derivations below. The key features of the Mellin transform for us are the following elementary relations; see <cit.> for their form without the Γ's in (<ref>).

	
	Let α,β∈, ν∈^N, 1≤ i ≤ N and f,g_+^N⟶. Writing i for the i-th unit vector, the (twisted) Mellin transform has the following properties:
	
		
  * Linearity: 
		α f+β g(ν)
			=αf (ν) + βg(ν),

		
  * Multiplication:
		x_i f (ν)
			=ν_i f (ν+i) and

		
  * Differentiation:
		∂_i f(ν)
			=- f (ν-i).
	


	The linearity is immediate from (<ref>) and the functional equation Γ(ν_i+1) = ν_iΓ(ν_i) provides the multiplication rule [x_i^ν_i-1/Γ(ν_i) ] x_i = ν_i x_i^ν_i/Γ(ν_i+1). The differentiation rule is a consequence of integration by parts,
	
    ∫_0^∞x_i^ν_i-1 x_i/Γ(ν_i)∂_i f
    		= [ x_i^ν_i-1/Γ(ν_i) f ]_x_i=0^∞
    		- ∫_0^∞x_i^ν_i-2 x_i/Γ(ν_i-1) f,

	because the boundary terms vanish inside the convergence domain of both integrals. This just says that if lim_x_i→ 0 (x_i^ν_i-ϵ-1 f) is finite for some ϵ>0, then lim_x_i→ 0 (x_i^ν_i-1 f) = 0 vanishes and the analogous argument applies to the upper bound x_i→∞.





 §.§ Operator algebras and annihilators




The Mellin transform relates differential operators acting on ^-/2 with operators that shift the indices ν of the Feynman integrals (ν). In this section we formalize this connection algebraically in the language of D-modules.
For the most part, we only need basic notions which we will introduce below, and point out <cit.> as a particularly accessible introduction to the subject.

	
	The Weyl algebra N in N variables x_1,…,x_N is the non-commutative algebra of polynomial differential operators

    N⟨
    		x_1,…,x_N,∂_1,…,∂_N  |   ∂_ix_j=x_j∂_i+δ_ij for all 1≤ i,j≤ N
    	⟩,

such that the commutators are [x_i,x_j]=[∂_i,∂_j]=0 and [∂_i, x_j] = δ_i,j (Kronecker delta).

Note that with the multi-index notations
x^α = x_1^α_1⋯ x_N^α_N
and
∂^β = ∂_1^β_1⋯∂_N^β_N,
every operator P∈N can be written uniquely in the form 

    P=∑_α,β
    		c_αβ x^α∂^β  with  α,β∈_0^N,

by commuting all derivatives to the right (only finitely many of the coefficients c_αβ∈ are non-zero).
Extending the coefficients c_αβ from  to polynomials [s] in a further, commuting variable s, we obtain the algebra
N[s] N_[s].
Later, we will also consider the case N_k N_ k of coefficients that are rational functions k(s).

	Given a polynomial f ∈[x], the N[s]-module [s,x,1/f]· f^s consists of elements of the form (p/f^k) · f^s (where p∈[s,x], k∈_0) with the N[s]-action 
	
    q ( p/f^k· f^s )
    		q p/f^k· f^s
    		,  ∂_i ( p/f^k· f^s )
    		f(∂_i p) + (s-k) p (∂_i f)/f^k+1· f^s

	for any polynomial q∈[s,x]. This is just the natural action by multiplication and differentiation, ∂_i ↦∂/(∂ x_i). We abbreviate f^s+k f^k · f^s for k ∈.
	The cyclic submodule generated by f^s is denoted as N[s] f^s.


	
	The s-parametric annihilators of a polynomial f in x_1,…,x_N are the elements of the (left) ideal of operators in N[s] whose action on f^s is zero:
	
    _N[s](f^s)
    		P∈N[s]P f^s=0.



	Given f∈[x_1,…,x_N], we always have the trivial annihilators
	
    f ∂_i - s (∂_i f) 
    		∈( f^s )
    		  for  
    		1 ≤ i ≤ N.


Note that an annihilator ideal is a module over N[s], that is, whenever P f^s=0, also (QP)f^s = Q(P f^s) = 0 for any operator Q∈N[s].
These ideals are studied in D-module theory <cit.> and in principle annihilators can be computed algorithmically with computer algebra systems such as  <cit.>. 
For the study of the Feynman integrals (<ref>), we set s=-/2 and f==+ is the polynomial from (<ref>). Via the Mellin transform, the elements of N[s] ^s are the integrands of shifts of the Feynman integral ^s.
Due to lemma <ref>, every annihilator
P ∈_N[s](^s)
corresponds to an identity of Feynman integrals with shifted indices ν.

	
	The algebra N of shift operators in N variables is defined by

    N⟨1,…,N,1,…,N  |  
    	[i,j]=[i,j]=0
    	 and 
    	[i,j] = δ_i,j⟩
    	.


This algebra is clearly isomorphic to the Weyl algebra N, since under the identifications i↔∂_i and j↔ x_j the commutation relations are identical. In fact, a different isomorphism is given by i↔ x_i and j↔ -∂_j, and it is this identification that corresponds to the Mellin transform (see lemma <ref>).
We therefore denote it by[
	The use of the symbol · for both the analytic Mellin transform (<ref>) and the isomorphism (<ref>) should not lead to any confusion. It is suggestive of, and justified by, corollary <ref>.
]

    ·N≅⟶N
    	,  
    	P ↦PP
    		x_i ↦i, ∂_i ↦ - i for all 1≤ i ≤ N
    	.


The conceptual difference between N and N is that we think of N as acting on functions f(x) by differentiation, whereas N acts on functions F(ν) of a different set ν=(ν_1,…,ν_N) of variables (the indices of Feynman integrals) by shifts of the argument and, in case of i, a multiplication with ν_i:

    (i F)(ν)   F(ν - i)
    	  and  
    	(i  F)(ν) ν_i F(ν + i)
    	.

These operators are used very frequently in the literature on IBP relations, as for example in <cit.>.
An important role is played by the operators

    iii,
    	  which act by simple multiplication:  
    	(i F)(ν) = ν_i F(ν)
    	.

Their commutation relations are

    [i, j]
    	=  iδ_i,j  and  
    	[i, j]
    	= - iδ_i,j
    	.

In terms of the N action (<ref>), we can rephrase the essence of lemma <ref> as

	
	The (twisted) Mellin transform (<ref>) is compatible with the actions of the algebras N and N under their identification (<ref>). In other words,
	
    P f^s
    		= P[ f^s]
    		  for all operators  
    		P ∈N[s]
    		.



	
	Every annihilator P ∈_N[]( ^s) of ^s=^-/2 yields a shift relation P∈N[s] N[s] of the Feynman integral  from (<ref>): P = 0.


	
	For the bubble graph in figure <ref> with =x_1+x_2-p^2 x_1 x_2 from example <ref>,

    (-p^2) x_1(s-x_1∂_1) + (s-x_1∂_1-x_2∂_2)
    	   ∈(^s)

	is easily checked to annihilate ^s. We therefore get the shift relation

    (s+1 +2)
    	= p^2 1 (s+1)
    	= p^2 (s+1+1)1
    	.

	According to (<ref>), this relation can also be written as
	
    (-p^2) ν_1 (ν_1+1,ν_2)
    		= - s+ν_1+ν_2/s+ν_1+1(ν_1,ν_2)
    		.


We prefer to work with the modified Feynman integral  from (<ref>), because it is directly related to the Mellin transform. However, it is straightforward to translate relations between  into relations for the actual Feynman integral .
Namely, if 
P
	=∑_α,β c_α,β x^α∂^β∈_N[s]( ^s ), we substitute (<ref>) to see

    0 = P
    	=∑_α,β c_α,β(∏_i,j=1^N ( i)^α_i (-j)^β_j) Γ(-s-)/Γ(-s)

and then recall from (<ref>) that = ∑_i ν_i +  s to conclude

    0 = Γ(-s)P/Γ(-s-) 
    	=∑_α,β c_α,βΓ(-s--α+β)/Γ(-s-)(∏_i,j=1^N ( i)^α_i (-j)^β_j)  
    	.

Notice that the fraction Γ(-s--α+β)/Γ(-s-) is a rational function in  (and thus in ν and s), due to the functional equation for Γ, since α and β are integers.

	Substituting (ν_1,ν_2)=(ν_1,ν_2) Γ(-2s-ν_1-ν_2)/Γ(-s) and (ν_1+1,ν_2) =(ν_1+1,ν_2) Γ(-2s-ν_1-1-ν_2)/Γ(-s) from (<ref>) into (<ref>) results in
	
    (-p^2) ν_1 (ν_1+1,ν_2) = (s+ν_1+ν_2)(2s+ν_1+ν_2+1)/s+ν_1+1(ν_1,ν_2)
    		,

	which also follows from the expression (<ref>) of (ν) in terms of Γ-functions.

Let us recapitulate these observations: By the Mellin transform (<ref>), every annihilator 
P ∈_N[s](^s)
gives rise to a linear relation P=0 of the rescaled Feynman integral  via a shift operator P∈N[s].
Also we noted that according to =  Γ(-s-)/Γ(-s) from (<ref>), such a relation is equivalent to a relation of the original Feynman integral  as in (<ref>).

On the other hand, if we are given a shift relation R = 0 where R∈N[s], then R=P corresponds to the differential operator P=[-1]R∈N[s] under the Mellin transform (<ref>). From the vanishing P ^s=0 we can conclude that this operator must be an annihilator,
P ∈_N[s](^s). This follows from

	
	Suppose that the (twisted) Mellin transform f^⋆(ν) f(ν) of f(x) from (<ref>) converges in a domain of the form a_i≤(ν_i) ≤ b_i for all 1≤ i ≤ N, where a,b ∈^N.
	Then its inverse is given by
	
    f(x)
    		= [-1]f^⋆(x)
    		= (∏_i=1^N 
    			∫_σ_i+ν_i Γ(ν_i)/ x_i^ν_i· 2π)
    		f^⋆(ν),
    		  where  
    		x ∈_+^N.

	This multiple integral along lines parallel to the imaginary axis converges for a_i≤σ_i ≤ b_i (1≤ i ≤ N) and does not depend on the concrete choice of σ_i.

So not only do we get relations for Feynman integrals from parametric annihilators of ^s, but in fact every relation of the form P=0 for a polynomial shift operator P∈N[s] does arise in this way.

	
	Let _N[s]() ⊆N[s] denote the N[s]-module of polynomial shift operators that annihilate a (rescaled) Feynman integral (<ref>).
	Then the Mellin transform (<ref>) restricts to a bijection between these relations and parametric annihilators:
	
    ·_N[s]( ^s ) 
    		⟶_N[s]( )
    		.


Instead of focussing on the annihilators themselves, we can also look at the N[s]-module N[s] ·^s N[s]/_N[s](^s) of the integrands and the N[s]-module N[s] ·N[s]/_N[s]() of all (shifted) Feynman integrals. The Mellin transform gives an isomorphism

    ·N[s]·^s ⟶N[s] ·
    	.







 §.§ On the correspondence to momentum space




In this section we first recall the integration by parts (IBP) relations for Feynman integrals that are derived in momentum space, following <cit.>. We then note that these provide a special set of parametric annihilators and discuss some open questions in regard of this comparison of IBPs in parametric and momentum space.


Since the denominators _a from (<ref>) are quadratic forms in the =+ momenta q=(q_1,…,q_)  (_1,…,_,_1,…,_), we can write them in the form

    _a
    	=∑_i,j∈_a^i,j s_i,j+λ_a

such that the coefficients _a^i,j and λ_a are independent of loop momenta and the pairs

    i,j1≤ i ≤ and 1≤ j ≤

label the = (+1)/2+ loop-momentum dependent scalar products 

    s_i,j q_iq_j = q_jq_i.

In order to express the IBP relations coming from momentum space in terms of the integrals (<ref>), we need to assume, for this and the following section, that we consider N= denominators such that the N× N square matrix  defined by (<ref>) is invertible.[
	For integrals associated to Feynman graphs, the number of edges is often less than . In this case, one augments the list of inverse propogators by an appropriate choice of additional quadratic forms in the loop momenta, called irreducible scalar products (ISPs), to achieve N=.
	See example <ref>.
]
We think of _a^i,j as the element of  in row a and column i,j and write _i,j^a for the entry in row i,j and column a of ^-1,
such that the inverse of (<ref>) can be written as

    s_i,j = ∑_a=1^N_i,j^a(_a-λ_a)
    	  for all  i,j∈
    	.

We are interested in relations of the Feynman integral  from (<ref>), that is,

    (ν_1,…,ν_N)
    	= (∏_j=1^∫[] _j/π^/2) f
    	  with the integrand  
    	f = ∏_i=1^N_i^-ν_i
    	.


	
	The momentum space IBP relations of (ν_1,…,ν_N) are those relations between scalar Feynman integrals that are obtained from Stokes' theorem
	
    (∏_n=1^∫[] _n) ij f
    		= 0,

	where the operators ij are defined in terms of the momenta[
		Recall that q_1,…, q_ denote the loop momenta
		whereas q_+1,…, q_ are the external momenta.
	] as
	
    ij∂/∂ q_i· q_j
    		=
    		∑_μ=1^∂/∂ q_i^μq_j^μ  for  
    		i∈1,…,,
    		j∈1,…,M.


The following, explicit form of these relations as difference equations is essentially due to Baikov <cit.>; see also Grozin <cit.>. For completeness, we include the proof in appendix <ref>.

	Given a set of N= denominators  such that the matrix  defined by (<ref>) is invertible,
	every momentum-space IBP relation can be written explicitly as
	
    ij(ν_1,… ,ν_N) = 0

	where ij denotes shift operators, indexed by 1≤ i ≤ and 1≤ j ≤, that are given by
	
    ij
    			dδ_ij-∑_a,b=1^N C_aj^bia( b-λ_b)
    			   for j≤ and
    
    			-∑_a,b=1^N C_aj^bia(b-λ_b)-∑_a=1^N∑_m=+1^A_a^i,m q_jq_ma   for j>.

	The coefficients C^bi_aj are defined as
	
    C_aj^bi∑_m=1^_a^{i,m}_m,j^b(1+δ_mi)    if j≤ and
    ∑_m=1^_a^{i,m}_m,j^b(1+δ_mi)    if j>.




	
	To every difference equation ij=0 from momentum space IBP, there corresponds a parametric annihilator ij∈_N[s]( ^s ) of the form
	
    ij    = dδ_ij+∑_a,b=1^N C_aj^bix_a(∂_b+λ_bH)
    		  for   i, j≤  and
    ij    = ∑_a,b=1^N C_aj^bix_a(∂_b+λ_bH)-∑_a=1^N∑_m=+1^_a^i,mq_jq_mx_aH
    			  for  
    			i≤<j,

	where
	H (+1)/2 + ∑_c=1^Nx_c∂_c.


	First recall the rescaling (<ref>) between the Feynman integral  and the Mellin transform  of ^s. As we saw in (<ref>), this means that
	
    Γ(-s-)/Γ(-s)a
    		= Γ(-s-) a/Γ(-s-)
    		= ( -s--1 ) a
    		= a (-s-) ,

	and so if we substitute (<ref>) into ij=0 for the operators from (<ref>), then apart from the substitution ab↦ x_a(-∂_b) which does not change , the remaining terms with shifts a do increment  by one and thus acquire an additional factor of
	
    -s-
    		=(+1)(-s) - ∑_i i↦
    		(+1)(-s) + ∑_c=1^N x_c ∂_c
    		= H.

	This proves that ij = 0 for the operators in (<ref>) and (<ref>) and theorem <ref> concludes the proof.


Note that the proof of the identity ij^s = 0 given in corollary <ref> rests on the inverse Mellin transform. An alternative, direct algebraic proof is given in appendix <ref>.[
	For an algebraic proof of the analogous statement in the Baikov representation, see <cit.>.
]

	
	By  we denote the left N[s]-module generated by the annihilators ij from corollary <ref>, corresponding to the momentum space IBP identities:
	
    ∑_i,jN[s] ·ij  ⊆  _N[s]( ^s )
    		.


Since the ij are first order differential operators, we have the inclusions

    ⊆^1_N[s](^s)
    	⊆_N[s](^s),

where ^1 denotes the N[s]-module generated by all first order annihilators. Note that for a generic polynomial , one would not expect that all of its annihilators can be obtained from linear ones (^1⊊). It is therefore interesting that we observe the equality ^1= in all cases of Feynman integrals that we checked.

	For a Feynman integral with a complete set of irreducible scalar products, is the second inclusion in (<ref>) an equality? In other words, are the s-parametric annihilators of Lee-Pomeransky polynomials  linearly generated?

Regarding the first inclusion in (<ref>), we do know that it is strict (see the example in appendix <ref>). 
However, it seems that  does provide all identities once we enlarge the coefficients to rational functions in the dimension s=-/2 and the indices ν_e.
Let us write θ=(θ_1,…,θ_N) and θ_e  x_e ∂_e such that ν_e=-θ_e. 

	Given any annihilator P∈_N[s](^s), does there exist a polynomial q ∈[s,θ] such that q P ∈? In other words, does
	
    (s,θ) _[s,θ]_N[s](^s)
    		=
    		(s,θ) _[s,θ]  hold?


To test this conjecture, we should take all known shift relations for Feynman integrals, and check if they can be realized as elements of  (after localizing at (s,θ)). In the remainder of this section, we will address such a relation, namely the one originating from the well-known dimension shifts.




 §.§ Dimension shifts



The representation = e^-/·^s from (<ref>) shows, through corollary <ref>, that

    ()
    
    	= (+2)

where = (1,…,N) is obtained from the polynomial (x_1,…,x_N) by substituting x_i ↦i.
This raising dimension shift was pointed out by Tarasov <cit.>,
[
	Tarasov considered the special case where all inverse propagators are of the form _e = k_e^2 - m_e^2 and hence  is just the graph (first Symanzik) polynomial from (<ref>). 
]
and had been observed before in special cases <cit.>.
For ()=[Γ(/2-)/Γ(/2)] (), the relation takes the form

    ()
    	= s/s+(1,…,N) (+2)
    	.

At the same time, the representation =^s implies also that

    () 
    	= (1,…,N) (+2)
    	= (+2)
    	.


	The equality of (<ref>) and (<ref>) implies, via the Mellin transform, that

    H(s)  + s 
    	= -(s-∑_a x_a ∂_a +  s)  + s ∈(^s-1)
    	.

Indeed, 
H(s) ^s
	= s( + (+1)) ^s-1-s(+1)^s
	= -s^s-1
follows from the homogeneity of  and , see (<ref>).[
	In fact, 
H(s) + s = ∑_e x_e ( ∂_e - (s-1) (∂_e ) )
follows from the trivial annihilators (<ref>) of ^s-1.
]

A lowering dimension shift, expressing (+2) in terms of (), corresponds to a Bernstein-Sato operator of  under the Mellin transform ()=^-/2:[
	Tkachov proposed in <cit.> to use a generalization of (<ref>) to several polynomials (the individual Symanzik polynomials  and , instead of =+), which in the physics literature is referred to as Bernstein-Tkachov theorem. However, this result is in fact due to Sabbah <cit.> (see also <cit.>).
]

	A Bernstein-Sato operator P(s)∈N[s] for a non-constant polynomial f is a polynomial differential operator such that there exists a polynomial b(s) ∈[s] with

    P(s) f^s+1 = b(s) f^s
    	.


Such operators always exist, and the Bernstein-Sato polynomial is the unique monic polynomial b(s) of smallest degree for which (<ref>) has a non-zero solution <cit.>. Given a solution of (<ref>) for f=, we get a lowering dimension shift relation:

    (+2)
    	= 1/b(s-1)P(s-1)()
    	.


	If we allow the coefficients to be rational functions k=(s), every integral in +2k dimensions can be written as an integral in  dimensions. In other words, the multiplication with f is invertible on N_k f^s. Put still differently, f^s generates the full module
	
    N_k · f^s
    		= k[x,1/f] · f^s
    		.



	By (<ref>), f^s-n = P(s-n)/b(s-n)⋯P(s-1)/b(s-1) f^s ∈N_k · f^s for all n∈.

In general, computing a Bernstein operator is not at all trivial. 
But in the case of a complete set of irreducible scalar products (N= and  is invertible), an explicit formula for the lowering dimension shift follows from Baikov's representation <cit.> of Feynman integrals. We use the form (<ref>) given by Lee in <cit.>:

Recall that (q_1,…, q_)=(_1,…,_,_1,…,_E) denotes the combined loop- and external momenta (= +). We introduce the Gram determinants

    _n(s) [ q_n · q_n         ⋯  q_n · q_;         ⋮         ⋱         ⋮;   q_· q_n         ⋯    q_· q_;           ]
    	= ( s_i,j)_n ≤ i,j≤

and remark that _+1(s) =  (_i ·_j)_1≤ i,j≤ depends only on the external momenta.
Furthermore, note that _1(s) is a polynomial in the scalar products s_i,j=q_i · q_j. By (<ref>), we can think of it also as a polynomial in the denominators .

	The Baikov polynomial (y) ∈[y_1,…,y_N] is the polynomial defined by (_1,…,_N) = _1(s).


	
	The Feynman integral (<ref>) can be written as
	
    ()
    	= c ·π^-/2-(-1)/4/Γ( --+1/2) ⋯Γ( -/2)·(-1)^/2/^(-E-1)/2( ∏_e=1^N∫ y_e/y_e^ν_e)
    	·{(y) }^(-N-1)/2

	where c∈ is a rational constant and the Baikov polynomial (y) has degree at most =+.
	The contour of integration in (<ref>) is such that  vanishes on its boundary.


We include the proof in appendix <ref>. For us, the interesting feature of this alternative formula is that the dimension appears with a positive sign in the exponent of the integrand. We can therefore directly read off

	A Feynman integral in +2 dimensions can be expressed as an integral in  dimensions by[
		The first fraction can also be written as 2^/(--+1 )_ in terms of the Pochhammer symbol (raising factorial) a_ = a (a+1) ⋯ (a+-1).]
	
    (+2)
    		= 
    		(-1)^/(--+1/2)
    			⋯(-/2)
    		(1,…,N)/()
    		.



	According to (<ref>), (+2) is obtained by multiplying the integrand of () with (-1)^(y)/ and adjusting the Γ-factors in the prefactor as Γ(+2-/2) = -/2Γ( -/2) and so on. Multiplying the integrand of the Baikov representation with y_e is equivalent to decrementing ν_e, hence the multiplication of the integrand by (y) can be written as the action of (1,…,N) on the integral.


The equation (<ref>) can be thought of as a Bernstein equation for (), or, equivalently, as a special type of integral relation: Combining (<ref>) with the raising dimension shift (<ref>), we find that

    {(1,…,N)/(1,…,N)
    		-
    		  -+2/2⋯-+(+1)/2}()
    	=0
    	.


We ask if this annihilator is contained in ; in other words, whether the lowering dimension shift relation (<ref>) is a consequence of the momentum space IBP identities. This is what we will establish in the following proposition <ref>.
First let us write the Baikov polynomial (y) explicitly: The block decomposition (s_i,j)_1≤ i,j≤ = ([  V  B; B^  G;    ]) with V=(_i·_j)_i,j ≤, B=(_i·_j)_i≤,j≤ and G=(_i·_j)_i,j≤ shows that

    (y)/
    	=  Q(y)
    	  where  
    	Q()  V - B G^-1 B^

is an × matrix whose entries are quadratic in the denominators. By (<ref>),

    Q_i,j
    	= ^a_i,j( _a - λ_a)
    	- ^a_i,r( _a - λ_a) 
    	  G^-1_r,s^b_j,s( _b - λ_b)

where we suppress the explicit summation signs over a=1,…,N in the first summand and over a,b=1,…,N and r,s=+1,…, in the second summand.

	
	The annihilator (<ref>), corresponding to the lowering dimension shift (<ref>), is contained in the ideal of shift operators generated by the momentum space IBP's from proposition <ref>:
	
    Q(1,…,N)
    		·(1,…,N)
    		- ∏_j=2^+1(+j-)/2∈∑_i,jN[] ·ij.



	Let us abbreviate (1,…,N) from (<ref>) such that =Λ and similarly Q̃ Q(1,…,N) for the matrix (<ref>). Using (<ref>) and (<ref>), we compute
	
    Q̃_i,jc   =
    			^a_i,jac 
    			- G^-1_r,s^a_i,r^b_j,s(
    				(a-λ_a) b-λ_bc
    				+ a-λ_ac (b-λ_b)
    			)
    		
       = -^c_i,j + G^-1_r,s(
    			^a_i,r^c_j,s (a - λ_a)
    			+^c_i,r^b_j,s (b - λ_b)
    		)
    		.

	Contracting with the matrix _c^k,l (for k,l≤) by summing over c, we conclude that
	
    Q̃_i,j_c^k,lc
    		= -δ_i,j,k,l + G^-1_r,s(
    			^a_i,rδ_j,s,k,l (a - λ_a)
    			+δ_i,r,k,l^b_j,s (b - λ_b)
    		)
    		.

	Note that the indices r and s take values >, whereas i and j are ≤. Hence δ_j,s,k,l=δ_i,r,k,l = 0. So, recalling (<ref>), we finally arrive at
	
    Q̃_i,j_k,l
    		= 1+δ_k,l/2δ_i,j,k,l
    		= δ_i,kδ_j,l + δ_i,lδ_j,k/2
    		.

	Recall (<ref>), that (x) = (x), such that 
	
    Q(1,…,N) ·(1,…,N)
    		= Q̃·.

	We can now invoke an identity of Turnbull <cit.>, see also <cit.> for a combinatorial and <cit.> for an algebraic proof, which relates this product of determinants to a determinant of the product Q̃·. This is non-trivial, because the elements of these two matrices do not commute, according to (<ref>).
	Turnbull's identity, as stated in <cit.>, applies precisely to this kind of very mild non-commutativity (<ref>) and states that
	
    Q̃·
    		= ( Q̃· + Q_col)
    		,
    		  where  
    		(Q_col)_i,j --i/2δ_i,j

	is a simple diagonal matrix and  denotes the column-ordered determinant
	
    A
    		∑_σ∈ S_N(σ) A_σ(1),1⋯ A_σ(N),N
    		.

	So let us now compute the entries of the product of Q̃ from (<ref>) with . Firstly,
	
    ^b_j,s(b-λ_b) _j,k
    		= -1/2^b_j,s(1+δ_j,k)^j,k_c (b-λ_b) c
    		= -1/2 C_cs^bk{c(b-λ_b)-δ_b,c}

	according to (<ref>). Note that C_bs^bk = ∑_b,j (1+δ_j,k) _b^j,k^b_j,s = ∑_j (1+δ_j,k) δ_j,k,j,s = 0 due to s>≥ k. So we can rewrite, due to (<ref>),
	
    ^b_j,s(b-λ_b) _j,k
    		=
    		1/2ks 
    		+ 1/2∑_m>^k,m_b b (q_s · q_m)
    		.

	Note that q_s · q_m = G_s,m such that contraction of the second summand with G^-1_r,s produces δ_r,m. So the sum over m collapses, and up to the term with ks, Q̃_i,j_j,k is
	
    -1+δ_j,k/2^a_i,j (a - λ_a) _c^j,kc
    		-1/2^a_i,r(a - λ_a) ^k,r_b b
    		=-1/2 C^ak_bi (a-λ_a) b
    
    		=1/2{ki -δ^k_i + C^ak_ai}
    		=1/2{ki -δ^k_i + (+1) δ^k_i },

	where the term
	C_ai^ak
	= ∑_a,j (1+δ_j,k) _a^j,k^a_j,i
	= ∑_j (1+δ_j,k) δ_j,k,j,i
	=δ_i^k (+1)
	comes from commuting a with b.
	Putting our results together, we arrive at
	
    Q̃_i,j_j,k
    		= 1/2{ 
    			(+1-)δ^k_i
    			+ ki 
    			- ^a_i,r (a-λ_a) G^-1_r,sks}
    		.

	So if we ignore all terms that lie in the (left) ideal generated by the momentum space (shift) operators ij, the column determinant (<ref>) of the matrix Q̃· + Q_col from (<ref>) can be replaced by an ordinary determinant B of the diagonal matrix
	
    B_i,j 
    		= δ_i,j/2(+1- -+i )
    		= δ_i,j/2(+1-+i ),

	such that indeed we conclude with the result that
	
    Q̃·≡∏_i=1^+1-+i/2∑_i,jN[] ·ij.


The Mellin transform ()=^s e^-/ identifies (<ref>) with the annihilator

    { Q(-∂)  ·  - b̃(s) }∙^s e^-/
    	= 0,
    	  where  b̃(s) ∏_j=2^+1(s++j/2)
    	.

To phrase this in terms of () = ^s = () ·Γ(-s-)/Γ(-s), we can use ^s = -H(s+1) ^s+1/(s+1) from remark <ref> to conclude that

    Γ(H(s)) · Q(-∂) ·1/Γ(H(s)--1)·^s+1 
    	= -(s+1)b̃(s) ^s,

where H(s)=[-1]-s- = -s(+1)+∑_i=1^N θ_i. Recall from (<ref>) that the left-hand side of (<ref>) can be written, in terms of the homogeneous components Q_r (with degree r) of Q(-∂) = ∑_r Q_r, as

    ∑_r Γ(H(s))/Γ(H(s)-L-1+r)  Q_r
    	= ∑_r ≤+1[ ∏_i=1^+1-r (H(s)-i) ] Q_r
    	+ ∑_r > +1[ ∏_i=0^r--11/H(s)+i] Q_r
    	.

If r ≤ + 1, this is a polynomial differential operator, and we thus obtained an explicit Bernstein-Sato operator as in definition <ref>.

	If the degree of the Baikov polynomial (y) is not more than +1, then the Bernstein-Sato polynomial b(s) of the Lee-Pomeransky polynomial  is a divisor of (s+1)b̃(s). In particular, all roots of b(s)/(s+1) are simple and at half-integers.

Note that (y) ≤min2, =  + min, by definition <ref> and equation (<ref>), so in particular, the corollary applies to all propagator graphs (=1) and to all graphs with one loop (=1).






§ EULER CHARACTERISTIC AS NUMBER OF MASTER INTEGRALS




Here we will show, using the theory of Loeser and Sabbah <cit.>, that the number of master integrals equals the Euler characteristic of the complement of the hypersurface defined by =0 inside the torus ^N (we write = ∖0 for the multiplicative group and  for the affine line).
For a full understanding of this section, some knowledge of basic D-module theory is indispensable; but we tried to include sufficient detail for the main ideas to become clear to non-experts as well. In particular, we will give self-contained proofs that only use D-module theory at the level of <cit.>.

	By V_ we denote the vector space of all Feynman integrals associated to , over the field (s,ν) (s,ν_1,…,ν_N) of rational functions (in the dimension and indices).
	More precisely, with _^s,
	
    V_∑_n ∈^N(s,ν) ·_(ν+n)
    		=
    		(s,ν) _[s,ν]( N[s] ·_)
    		.

	The number of master integrals is the dimension of this vector space:
	
    _(s,ν) V_
    		.


Note that this is the same as the dimension of the space ∑_n (s,ν) _(ν+n) of Feynman integrals (<ref>), because the ratios _(ν+n)/_(ν+n) = Γ(-s)/Γ(-s--n) with n=n_1+…+n_N are all related by a rational function in (s,ν), see (<ref>).

	The phrase “master integrals” is used in the existing physics literature to denote various different quantities, and none of those notions coincides exactly with ours.
	The main sources for discrepancies are:
	
		
  * Almost always the integrals are considered only for integer indices ν∈^N, instead of as functions of arbitrary indices. In this setting, integrals with at least one ν_e=0 can be identified with quotient graphs (“subtopologies”) and are often discarded from the counting of master integrals.
		
  * We only discuss relations of integrals that are expressible as linear shift operators acting on a single integral. 
	This setup cannot account for relations of integrals of different graphs (with some fixed values of the indices), as for example discussed in <cit.>.
	It also excludes symmetry relations, which are represented by permutations of the indices ν_e.
	
  * Some authors do not count integrals if they can be expressed in terms of Γ-functions or products of simpler integrals, for example <cit.>.
	
	Taking care of these subtleties, we will demonstrate in section <ref> that our definition gives results that do match the counting of master integrals obtained by other methods.


A fundamental result for methods of integration by parts reduction is that the number of master integrals is finite. This was proven in <cit.> for the case of integer indices ν∈^N, using the momentum space representation. Below we will show that this result holds much more generally, for unconstrained ν, and that it becomes a very natural statement once it is viewed through the parametric representation.
Notably, it remains true for Mellin transforms ^s of arbitrary polynomials —the fact that  comes from a (Feynman) graph is completely irrelevant for this section.

Recall that, by the Mellin transform, we can rephrase statements about integrals in terms of the parametric integrands. In line with (<ref>) and (<ref>), we can rewrite (<ref>) as

    = _(s,θ)(
    		(s,θ)
    		_[s,θ]N[s] ·^s
    	)
    	,

where [s,θ] = [s,θ_1,…,θ_N] denotes the polynomials in the dimension s=-/2 and the operators θ_e  x_e ∂_e = [-1]-ν_e, and F (s,θ) stands for their fraction field (the rational functions in these variables). Since F contains k (s), we can equivalently work over this base field throughout and write

    = _F (F_R)

in terms of R  k[θ] and the module = N_k ·^s over the Weyl algebra N_k N_ k = N[s] _[s] k over the field k=(s).
Crucially, N_k ·^s is a holonomic N_k-module, which is a fundamental result due to Bernstein <cit.>.

Holonomic modules are, in a precise sense, the most constrained, and behave in many ways like finite-dimensional vector spaces. For example, sub- and quotient modules, direct and inverse images of holonomic modules are again holonomic <cit.>, and holonomic modules in zero variables are precisely the finite-dimensional vector spaces. The holonomicity of the parametric integrand was already exploited in <cit.> to show that Feynman integrals fulfill a holonomic system of differential equations, and it is also a key ingredient in the proof in <cit.>.

The number defined in (<ref>) has been studied by Loeser and Sabbah <cit.> in a slightly different setting, namely for holonomic modules over the algebra 

    N_k
    	 k[x_1^± 1,…,x_N^± 1] ⟨∂_1,…,∂_N ⟩
    	= k[x^± 1] _k[x]N_k 
    	= N_k [x^-1]

of linear differential operators on the torus [k]^N. Note that N_k is just the localization of N_k at the coordinate hyperplanes x_i=0; that is, the coefficients of the derivations are extended from polynomials (^N_k)=k[x] to rational functions ([k]^N)=k[x^± 1] whose denominator is a monomial in the coordinates x_i. Equivalently, we can also view N_k= ^∗N_k as the pull-back under the (open) inclusion

    [k]^N ^N_k
    	.

The pull-back along  turns every N_k-module  into a N_k-module ^∗, namely the localization ^∗ = k[x^± 1] _k[x] = [x^-1]. Importantly, if  is holonomic, so is its pull-back ^∗. The starting point for this section is

	
	Let  denote a holonomic N_k-module. Then F _R is a finite-dimensional vector space over F. Moreover, its dimension is given by the Euler characteristic _F (F _R ) = χ(^∗).

In appendix <ref>, we provide a self-contained proof of this crucial theorem, simpler and more explicit than in <cit.>.
For now, let us content ourselves with reducing it to the known situation on the torus.

	We can invoke _F(F _R ^∗) = χ(^∗) < ∞ from <cit.>. To conclude, we just need to note that F _R and F _R [x^-1] = F _R ^∗ are isomorphic vector spaces (over F). This is clear since each coordinate x_i is invertible after localizing at F: due to ∂_i x_i = 1+x_i ∂_i, we find that (1+x_i ∂_i)^-1∂_i ∈ F_R N_k is an inverse to 1 x_i.

This result not only implies the mere finiteness of the number of master integrals, but in addition gives a formula for this number—it is the Euler characteristic, given by

    χ( ' )
    	χ(('))
    	= ∑_i (-1)^i _k H^i ( ( ')  )
    	,

of the algebraic de Rham complex of '^∗. This is the complex

    (')
    	( Ω^∙_[k]^N⊗_([k]^N)'[N], )

of '-valued differential forms on the torus [k]^N, with the connection (ω m) = ω m + ∑_i=1^N ( x_i ∧ω) ∂_i m. Note that the r-forms ω are shifted to sit in degree r-N of the complex, which is thus supported in degrees between -N and 0; hence (<ref>) is a finite sum over 0 ≤ i ≤ N. The extremal cohomolgy groups are easily identified as

    H^-N( (') )
    	= ⋂_i=1^N ∂_i
    	  and  
    	H^0( (') )
    	. ' / ∑_i=1^N ∂_i ' .
    	= π_∗'
    	,

with the latter also known as push-forward of ' under the projection π[k]^N _k^0 to the point.
Since holonomicity is preserved under direct images, we conclude that _k H^0( (') ) is finite.[
	Recall that a holonomic module over the point ^0_k is the same as a finite-dimensional k-vector space.
]
In fact, the same is true for the other de Rham cohomology groups, which shows that (<ref>) is indeed well-defined.[
	The de Rham complex (N_k) is a resolution of k[x] by free N_k-modules, such that H^∙ ((')) are the (left) derived functors of π_∗'=H^0((')). In the language of derived categories, saying that π_∗' is holonomic actually means precisely that (') is a complex with cohomology groups that are holonomic modules over the point—that is, finite-dimensional vector spaces over k.
]

Since we are interested in Feynman integrals, we consider the special case where the N_k-module  is simply =N_k ·^s from definition <ref>. Its elements can be written uniquely in the form h·^s, where h ∈ k[x,^-1], such that N_k ·^s  k[x,^-1] by (<ref>) are isomorphic as k[x]-modules: x_i (h ^s) = (x_i h) ^s.
The action (<ref>) of the derivatives, however, is twisted by a term proportional to s:
∂_i (h ^s) = ^s (∂_i h + sh(∂_i )/).
Despite this twist, we find that the Euler characteristic stays the same:

	Let ∈[x_1,…,x_N] be a polynomial and set k=(s). Then the Euler characteristics of the algebraic de Rham complexes of the holonomic N_k-module ^∗N_k ^s and the holonomic N_-module [x^± 1,^-1]=([k]^N ∖()) coincide:
	
    χ(^∗N_k^s)
    		= χ([x^± 1,^-1])
    		.


In particular, we can dispose of the parameter s completely and compute with the algebraic de Rham complex of [x^± 1,^-1], which is the ring of regular functions of the complement of the hypersurface ()=x(x)=0 in the torus ^N. 
Combining theorem <ref> with proposition <ref>, we thus obtain our main result:

	The number of master integrals of an integral family with N denominators is
	
    = χ([x^± 1,^-1] ),

	the Euler characteristic of the algebraic de Rham complex of the complement of the hypersurface x_1⋯ x_N ·=0 inside the affine plane ^N.
	Via Grothendieck's comparison isomorphism, this is the same as the topological Euler characteristic, up to a sign:[
		This sign arises from the shift by N in the definition (<ref>) of the de Rham complex .
	]
	
    = (-1)^N χ(^N ∖x_1⋯ x_N ·=0)
    		= (-1)^N χ(^N ∖=0)
    		.



	We stress that this geometric interpretation of the number of master integrals is valid for dimensionally regulated Feynman integrals, that is, we consider them as meromorphic functions in  (and ν). This is reflected in our treatment of s=-/2 as a symbolic parameter.
	
	If, instead, one specializes to a fixed dimension like =2 (s=-1) or =4 (s=-2), then (<ref>) is no longer true in general.[
		It fails precisely if, for some r ∈, s-r is a zero of the Bernstein-Sato polynomial of .
	]
	It can thus happen that N·^s ⊊[x,^-1] is a proper subalgebra (note k=(s)=). While theorem <ref> still applies and relates the number of master integrals in a fixed dimension to the Euler characteristic of N·^s, this is not always equal to the topological Euler characteristic (<ref>).
	This is expected, since the number of master integrals is known to be different in fixed dimensions <cit.>.


	Given an N_-module  and a polynomial f ∈[x], set ' [f^-1] and consider the N_-module 'f^s formed by products of f^s with elements m∈'[s] ' _[s]. As vector spaces, ' f^s '[s] via mf^s ↦ m, but the N_ action on 'f^s has twisted derivatives to take into account the factor f^s:
	
    x_i ∙ mf^s  x_i m f^s
    		  and  ∂_i ∙ m f^s
    		{ (∂_i m) + s m ∂_i f/f} f^s
    		.

	Following Malgrange <cit.>, we introduce the action of a further variable t by setting
	
    t ∙ m(s) f^s
    		 m(s+1) f^s+1  and  ∂_t ∙ m(s) f^s
    		 -s m(s-1) f^s-1
    		,

	where we use the intuitive abbreviation f^s+r f^r · f^s for r∈.
	One easily verifies [∂_t,t] = 1 and [∂_i,∂_t]=[∂_i,t]=[x_i,∂_t]=[x_i,t] = 0, such that ' f^s becomes an N+1_-module in the N+1 variables (x_1,…,x_N,t). Note that ∂_t t = -s, so
	
    ' f^s/∂_t ' f^s
    		= ' f^s/s t^-1' f^s
    		= ' f^s/s ' f^s'

	is an isomorphism of N_-modules.[
		This is also clear from the fact that ' f^s/(∂_t ' f^s) = π_∗ (' f^s) is the push-forward of ' f^s under the projection π^N+1⟶^N forgetting the last coordinate. Namely, since ' f^s = F_∗' as we discuss below, π_∗ (F_∗') = (π∘ F)_∗' = 𝕀_∗' = '.
	]
	Since ∂_t is injective on ' f^s (it raises the degree in s), the de Rham complex ('f^s) is quasi-isomorphic to (' f^s/∂_t ' f^s) = ('), see corollary <ref>. So we can conclude the equality
	
    χ(' f^s)
    		= χ(')
    		,

	once we assume that  is holonomic to ensure that these Euler characteristics are well defined. Indeed, the holonomicity of ' and ' f^s holds because
	
		
  * '=j^∗ is the pull-back of  under the inclusion j^N_∖f=0^N_,
		
  * ' f^s = F_∗' is the push-forward of ' under the closed embedding F^N_^N+1_ which sends x to (x, f(x)).[
		It follows from (<ref>) that 'f^s=⊕_n ≥ 0∂_t^n '  F_∗ as [x]-modules, since ∂^n_t '=∂^n_t t^n ' ≡ s^n '  s^<n'. Furthermore, the derivatives act on F_∗ by
		∂_i ∙ m(s)
		= (∂_i - (∂_i f) ∂_t)m(s)
		= (∂_i +s (∂_i f) t^-1)m(s)
		= (∂_i +s (∂_i f)/f)m(s-1)
		in accordance with (<ref>).
	]
	
	Alternatively, the filtration Γ_j ' f^s  f^-j∑_i≤ j s^i Γ_2j( f)-i induced by any good filtration Γ_∙ on  directly shows the holonomicity of ' f^s, since its dimension grows like j^N+1 for large j.
	We now invoke the theory of Loeser-Sabbah to deduce that
	
    χ('f^s)
    		= _(θ,t∂_t)'f^s (θ, t∂_t)
    		= _k(θ) (θ)
    		= χ()

	where 'f^s (t∂_t) denotes the algebraic Mellin transform (<ref>) of 'f^s with respect to the coordinate t. But note that, according to (<ref>), localizing at t∂_t = -s-1 just extends the coefficients to k=(s). So = ' _(s)f^s = _ k f^s is just the holonomic N_k-module on the left-hand side of (<ref>), because, over k, f is invertible by corollary <ref>.
	We have proven χ(_ k f^s) = χ([f^-1]), and the special case of =[x^± 1] with f = proves the claim.


	More abstractly, Proposition <ref> can also be seen as an application of the theory of characteristic cycles <cit.>: 
It is known that the Euler characteristic only depends on the characteristic cycle of a N_k-module, which follows from the Dubson-Kashiwara formula <cit.>.
Therefore it is sufficient to show that the N_k-modules k[x^± 1,^-1] and k[x^± 1] ^s have the same characteristic cycles, via <cit.>. This follows from the fact that these modules are identical up to the twist by the isomorphism ∂_i ↦∂_i + s (∂_i )/ of N_k[^-1].







 §.§ No master integrals



Corollary <ref> shows in particular that there are no master integrals, =0, precisely when the Euler characteristic χ(^N∖(f)) χ([x^± 1,f^-1]) = χ( (^∗)^N ∖(f)) vanishes for f=. For example, this happens if f is homogeneous in a generalized sense: Suppose we can find λ_0,…,λ_N ∈, not all zero, such that

    f(x_1 t^λ_1,…,x_N t^λ_N)
    	= t^λ_0 f(x_1,…,x_N)
    	  in  [x,t^± 1]
    	;

which is equivalent (apply ∂_t and set t=1) to the existence of a linear annihilator,

    P^λ_s∙ f^s = 0,
    	  of the form  
    	P^λ_s ∑_i=1^N λ_i θ_i - sλ_0
    	∈[s,θ] ∖0
    	.


	Given f∈[x_1,…,x_N], the Mellin transform _(s)[θ](s,θ) of N_k· f^s is zero if, and only if, f^s is annihilated by a polynomial in the Euler operators θ:
	
    _N_k(f^s) ∩[s,θ] ≠0.



	Clearly, =0 requires f^s to be mapped to zero in the localization  of  at [s,θ]∖0, and therefore the existence of a non-zero polynomial P(θ,s)∈[s,θ] with P(θ,s)∙ f^s =0. Conversely, given such an operator, its shifts P(θ-α,s+r) by (r,α)∈^1+N annihilate the elements x^α· f^s+r, which are therefore all mapped to zero in . By linearity, this proves =0, because every element of  can be written as g f^s+r h for some r ∈, h ∈[s,θ]∖0 and a Laurent polynomial g ∈ k[x^± 1].

In particular, the presence of a linear annihilator (<ref>) implies =0 and hence χ(^N∖(f))=0 via corollary <ref>. Note that we could equally phrase this in terms of the hypersurface (f) ⊂^N itself as χ((f))=0, because the Euler characteristics are related through χ((f))=-χ(^N∖(f)) (see section <ref>).

When f= comes from Feynman graph G (as in the next section), it is not difficult to see that the homogeneity (<ref>) occurs precisely when G has a tadpole.[
	A tadpole here means a proper subgraph γ⊊ G which shares only a single vertex with the rest of G and does not depend on masses or external momenta. In this case, _G=_γ_G/γ factorizes such that the variables x_i with i∈γ only appear in the homogeneous polynomial _γ of degree λ_0_γ. Thus we obtain (<ref>) by setting  λ_e = 1 if e∈γ and λ_e=0 otherwise.
]
If this is the case, the integrals from proposition <ref> do not converge for any values of s and ν. In fact, =0 dictates that the only value one can assign to f^s(ν) which is consistent with integration by parts relations is zero. This reasoning explains a common practice in Feynman integral calculations, namely that Feynman integrals associated to graphs with tadpoles are declared to vanish.

The purpose of this section is to show that the simple homogeneity condition (<ref>) is not only sufficient for a vanishing Mellin transform, but it is also necessary:

	Let f ∈[x_1,…,x_N] denote a polynomial. Then the hypersurface f=0 inside the torus ^N has vanishing Euler characteristic precisely when there are λ_0,…,λ_N ∈, not all zero, such that (<ref>) holds.

Geometrically, the homogeneity (<ref>) can be interpreted as follows: dividing by the greatest common divisor, we may assume that λ_0,…,λ_N are relatively prime. Thus we may extend (λ_1,…,λ_N) to a basis of the lattice ^N and hence construct a matrix A ∈N with first row A_1i = λ_i.
In the associated coordinates y, defined by

    x_i=∏_j=1^N y_j^A_ji  and  
    	y_i=∏_j=1^N x_j^A^-1_ji  where  
    	A^-1_ji( A^-1)_ji,

the polynomial f takes the form f(x)=y_1^λ_0 g(y̅) for some Laurent polynomial g ∈[y̅^±1] in the remaining variables y̅=(y_2,…,y_N). In particular, the hypersurface f=0=g=0 can be defined by an equation independent of the coordinate y_1.

	Let f ∈[x_1,…,x_N] denote a polynomial. Then (f) ⊂^N has Euler characteristic zero if and only if it is isomorphic to a product of  times a hypersurface g=0⊂^N-1.

To prove proposition <ref>, we will look at the Newton polytope f of f, which is defined as the convex hull of the exponents of monomials that appear in f:

    ∑_α∈^N c_α x^αα∈^Nc_α≠ 0⊂^N
    	.

Since every monomial x^α is an eigenvector of the operators (<ref>), P^λ_s(θ)∙ x^α = P^λ_s(α) x^α, it is annihilated by P^λ_1 exactly when α belongs to F_λP^λ_1(α)=0, the hyperplane F_λ= αα_1λ_1+…+α_N λ_N=λ_0.
In particular, 0=P^λ_s ∙ f^s = s f^s-1 P^λ_1 ∙ f is equivalent to the Newton polytope f⊂ F_λ being contained in that hyperplane. We can therefore reformulate the equivalent conditions (<ref>) and (<ref>) as

    f < N.

Such polytopes have zero N-dimensional volume, and we call them degenerate.


	We proceed by induction over the dimension N, and we will assume f to be non-constant (the proposition holds trivially for any constant f∈).
	In the case N=1, the variety (f) ⊂^∗ is a finite set and hence its Euler characteristic coincides with its cardinality. Therefore, χ((f))=0 if and only if f has no zero inside the torus. This is only possible if f is proportional to a monomial x_1^r; in particular f must be homogeneous and we are done.
	
	Now consider N>1 and assume that χ(^N∖(f))=χ((f))=0.
	Recall that (<ref>) is equivalent to degeneracy (<ref>) of f, so we only need to rule out the non-degenerate case. We achieve this by exploiting the hypothesis f=N to construct a linear annihilator P^λ_s of f^s, which implies (<ref>) in contradiction to the non-degeneracy of f.

	To start, we use lemma <ref> to find a polynomial 0≠ P(θ,s) ∈[s,θ] such that P(θ,s)∙ f^s = 0, and we choose one with minimal total degree in s and θ.
	Then pick an (N-1) dimensional face σ = f∩ F_λ, which we can write as the intersection of f with a hyperplane F_λ for some integers λ_0,…,λ_N such that f⊆αP^λ_1(α) ≤ 0.
	Under the rescaling (<ref>), all monomials of f=∑_α c_α x^α with α∈σ⊂ F_λ acquire a factor of t^λ_0, while the remaining monomials with α∈f∖σ come with a smaller exponent ∑_i=1^N α_i λ_i<λ_0 of t:
	
    f(x_1 t^λ_1,…,x_N t^λ_N)
    		= t^λ_0 f_σ(x) ( 1+t^-1),
    		  where  
    		f_σ(x) ∑_α∈^N ∩σ c_α x^α

	and t^-1 denotes a rational function in t^-1(x)[t^-1]. Note that P(θ,s)∙ f^s(x_i t^λ_i) = 0 is still zero, because the rescaling of x commutes with the Euler operators θ_i∙ h(x_i t^λ_i) = (θ_i ∙ h(x_i))|_x_i↦ x_i t^λ_i. Therefore, applying P(θ,s) to the s-th power of the right-hand side of (<ref>) and dividing by t^sλ_0 yields
	
    0 = P(θ,s) ∙ f_σ^s(x) ( 1+t^-1)^s
    		= P(θ,s)∙ f_σ^s + t^-1f_σ^s,

	where t^-1 on the right-hand side denotes a formal series in t^-1(x,s) [[t^-1]]. In particular, the coefficient of t^0 must vanish, and we conclude that P(θ,s) ∙ f_σ^s=0. Label the variables such that λ_N ≠ 0, then we can divide P(θ,s) by the linear form P^λ_s(θ,s) from (<ref>), as a polynomial in θ_N, to obtain a decomposition
	
    P(θ,s) = P(θ',0,s) + P^λ_s (θ,s) · Q(θ,s)

	for some polynomial Q(θ,s)∈[θ,s], such that the first summand depends only on θ'(θ_1,…,θ_N-1) and s.
	Since f_σ=σ⊂ F_λ is contained in the hyperplane F_λ=αP^λ_1(α)=0, we see P^λ_s ∙ f^s_σ = f^s-1 P^λ_1 ∙ f_σ = 0 and thus Q(θ,s) drops out in
	
    0 = P(θ,s) ∙ f^s_σ = P(θ',0,s) ∙ f^s_σ
    		= P(θ',0,s) ∙ g^s
    		,

	where g  f|_x_N=1∈[x_1,…,x_N-1] is a polynomial in less than N variables. If P(θ',0,s) were non-zero, lemma <ref> would show χ(^N-1∖(g))=0, such that we could apply our induction hypothesis to g and conclude that g is homogeneous in our generalized sense. We saw that this is equivalent to the degeneracy of g, which contradicts that gf_σ=σ is of dimension N-1.[
		Observe that g is the orthogonal projection of f_σ onto the coordinate hyperplane α_N=0. This projection restricts to an isomorphism between α_N=0 and F_λ (because λ_N ≠ 0), and therefore g=f_σ.
	]

	Therefore, P(θ',0,s) must be zero and we conclude that P(θ,s)=P^λ_s · Q has a linear factor P^λ_s(θ,s).[
		The existence of a linear annihilator could also be deduced from <cit.>.
	]
	Now set m Q(θ,s)∙ f^s, which is non-zero, because P(θ,s) was chosen as an annihilator of f^s of minimal degree. We may write this element in the form m=a· f^s+r for some r∈ and a Laurent polynomial a ∈(s)[x^±]. After multiplying with with a polynomial in [s], we may even assume 0 ≠ a ∈[s,x^± 1] with P^λ_s ∙ af^s+r = 0. Applying the Leibniz rule and dividing by af^s+r, we find
	
    0
    		
    		= P^λ_0 ∙ a/a - sλ_0 + (s+r) P^λ_0 ∙ f/f
    		.

	Since the degree of P^λ_0 ∙ a in s is at most the degree (in s) of a itself, this first summand on the right has a finite limit as s→ 0. We therefore must have a cancellation of the terms linear in s, P^λ_0 ∙ f = λ_0 f, which yields the sought-after P^λ_s ∙ f^s = 0.



	In summary, we showed that the following six conditions on a polynomial f∈[x_1,…,x_N] are equivalent:
	[(1)]
		
  * homogeneity (<ref>) of f,
		
  * existence (<ref>) of an annihilator of f^s linear in θ,
		
  * existence  (<ref>) of an annihilator of f^s polynomial in θ,
		
  * degeneracy (<ref>) of the Newton polytope f,
		
  * vanishing of the Euler characteristic χ(^N∖(f))=χ((f))=0 and
		
  * divisibility of (f) by  as stated in corollary <ref>.
	

To conclude, let us interpret our observation in the light of the well-known result, due to Kouchnirenko <cit.> and Khovanskii <cit.>, that relates the Euler characteristic to the volume of the Newton polytope:

	For almost all polynomials f ∈[x_1,…,x_N] with a fixed Newton polytope, we have χ(^N∖(f))=(-1)^N · N! ·f.
	

For polynomials f whose non-zero coefficients are sufficiently generic, proposition <ref> follows from theorem <ref>. Our proof shows that when f=0, the theorem applies without any constraints on the non-zero coefficients of f. In fact, this statement extends to the case when N!f=1, because it is known that (-1)^N·χ(^N∖(f)) is always bounded from above by N!f, for all f, leaving only the possibilities 0,1 for the signed Euler characteristic (-1)^N·χ(^N∖(f)).




 §.§ Graph polynomials



Our discussion so far applies to all integrals of the type (<ref>)—the defining data is thus the set =(_1,…,_N) of denominators, which is sometimes also called an integral family <cit.>.
The denominators can be arbitrary quadratic forms in the loop momenta; the decomposition (<ref>) then defines the associated polynomials ,  and  through (<ref>). In particular, the denominators do not have to be related to the momentum flow through a (Feynman) graph in any way.

However, we will from now on consider the most common case in applications: integrals associated to a Feynman graph with Feynman propagators. 


Given a connected Feynman graph G with N internal edges, +1 external legs and  loops, imposing momentum conservation at each vertex determines the momenta _e flowing through each edge e in terms of the  external and  loop momenta.[
	Momentum conservation implies that the sum p_1+⋯+p_+1=0 of the incoming momenta on all external legs vanishes; hence only  of them are independent.
]
The Symanzik polynomials _G and _G of the graph G are the polynomials  and  from (<ref>) for the set =(_1,…,_N) of inverse Feynman propagators,[
	The infinitesimal imaginary part  is irrelevant for our purpose of counting integrals and will be henceforth ignored.
]

    1/_e = 1/-_e^2+m_e^2-  (1≤ e ≤ N)
    	,

where m_e is the mass associated to the particle propagating along edge e. 
	The number G of master integrals of the Feynman graph G is defined in terms of (<ref>) as
	
    G_G  where  _G _G+_G
    		.


This class of integrals (using only the propagators in the graph) is sometimes referred to as scalar integrals and might appear to be insufficient for applications, since in general one needs to augment the inverse propagators by additional denominators, called irreducible scalar products (ISPs), in order to be able to express arbitrary numerators of the momentum space integrand in terms of the integrals (<ref>); see example <ref>.
Therefore, one might expect that, in order to count all these integrals, one ought to replace _G in (<ref>) by the polynomial associated to the full set of denominators, including the ISPs.

However, it is well-known since <cit.> that all such integrals with ISPs are in fact linear combinations of scalar integrals in higher dimensions +2k, for some k∈.
Furthermore, those can be written as scalar integrals in the original dimension  by corollary <ref>. Therefore, (<ref>) is the correct definition to count the number of master integrals of (any integral family determined by) a Feynman graph.

We can therefore invoke the following, well-known combinatorial formulas for the Symanzik polynomials <cit.>, which go back at least to <cit.>.

	The Symanzik polynomials of a graph G can be written as

    _G = ∑_T∏_e ∉ T x_e
    	  and  _G = _G ∑_e=1^N x_e m_e^2
    	     - ∑_F p_F^2 ∏_e ∉ T x_e
    	,

	where T runs over the spanning trees of G and F enumerates the spanning two-forests of G (p_F denotes the sum of all external momenta flowing into one of the components of F).

In particular it is well-known that for all connected graphs, _G ≠ 0 and thus  is invertible and (<ref>) makes sense.
In section <ref>, we will use these formulas to count the sunrise integrals.

	The graph polynomials of the bubble graph (figure <ref>) are, in general kinematics,
	
    = x_1 + x_2
    		  and   = (x_1 + x_2)(x_1 m_1^2 + x_2 m_2^2) - p^2 x_1 x_2
    		.



It is important to keep in mind that, even with a fixed graph, the number of master integrals will vary depending on the kinematical configuration—e.g. whether a propagator is massive or massless, or whether an external momentum is non-exceptional or sits on a specific value (like zero or various thresholds).
We will always explicitly state any assumptions on the kinematics, and hence stick with the simple notation (<ref>).




 §.§ The Grothendieck ring of varieties



Since we are from now on only interested in the Euler characteristic, we can simplify calculations by abstracting from the concrete variety () =0 to its class [] in the Grothendieck ring K_0(Var_). This ring is the free Abelian group generated by isomorphism classes [X] of

varieties over ,
modulo the inclusion-exclusion relation [X]=[X∖ Z] + [Z] for closed subvarieties Z ⊂ X. 
It is a unital ring for the product [X]·[Y] = [X × Y] with unit 1=[^0] given by the class of the point.
Crucially, the Euler characteristic factors through the Grothendieck ring, since it is compatible with these relations: χ(X) = χ(X∖ Z) + χ(Z) and χ(X× Y) = χ(X)·χ(Y).
The class = [^1] of the affine line is called Lefschetz motive and fulfils χ()=1.
For several polynomials P_1,…,P_n, we write (P_1,…,P_n) P_1=⋯=P_n=0.

If a variety is described by polynomials that are linear in one of the variables, we can eliminate this variable to reduce the ambient dimension.[
	Such linear reductions were first investigated by Stembridge in <cit.> and have led, via the c_2-invariant <cit.> of Schnetz, to the discovery of graph hypersurfaces that are not of mixed Tate type <cit.>.
]
Let us state such a relation explicitly, since our setting is slightly different than usual: For us, the natural ambient space is the torus ^N and not the affine plane ^N.


	Let A,B ∈[x_1,…,x_N-1] and consider the linear polynomial A+x_N B. Then

    [^N ∖(A+x_N B)]
    	=·[ ^N-1∖(A,B) ]
    	 -[ ^N-1∖(A) ]
    	 -[ ^N-1∖(B) ]

holds in the Grothendieck ring. In particular, the Euler characteristic is

    χ( ^N ∖(A+x_N B) )
    	=- χ( ^N-1∖(A· B) )
    	.



Consider the hypersurface ()⊂^N, defined by A+x_N B, under the projection π^N ⟶^N-1 that forgets the last coordinate x_N.

As long as AB≠ 0, the unique solution of =0 in the fibre is x_N=-A/B. If A=0, the solution x_N=0 is not in  and the fibre is empty, and it is also empty whenever B=0. The only exception to this emptiness is over the intersection A=B=0, where x_N is arbitrary and the fibre is the full . This fibration proves (<ref>); equivalently, we can write it via [(A· B)] = [(A)] + [(B)] - [(A,B)] and []= - 1 as

    [^N ∖(A+x_N B)]
    	=
    	[](
    		[ ^N-1∖(A) ]
    		+[^N-1∖(B) ]
    	)
    	-· [ ^N-1∖(A· B) ]
    	.

Applying the Euler characteristic proves (<ref>) due to χ(^1)=1 and χ()=0.


	Set x_N=1 and x_N=1. Then

    χ( ^N ∖ () )
    	   = χ(^N-1∖( ,) )
    	- χ( ^N-1∖( ) )
    	- χ( ^N-1∖( ) )
    	
    	
       = - χ( ^N-1∖( · ) )
    	.



	Recall that  and  are homogeneous of degrees  and +1, respectively (corollary <ref>). Since multiplication with x_N ∈ is invertible, we can rescale all variables x_i with i<N by x_N. This change of coordinates transforms  into x_N^ (+x_N ). Since x_N ≠ 0, this shows that
[^N ∖()]
= [^N ∖( + x_N )]
such that the claim is just a special case of lemma <ref>.


	Both graphs consisting of a pair of massless edges,
	
    G_series=[0.5]serieslegs  and  
    		G_parallel=[0.5]parallellegs,

	with the external momentum p such that p^2 ≠ 0, have a single master integral G=1.


	According to (<ref>), the graph polynomials of the graphs in (<ref>) are
	_series = 1,
	_series = - p^2 (x_1 + x_2),
	_parallel = x_1 + x_2,
	_parallel = -p^2 x_1 x_2 such that
	
    _series
    		= 1 - p^2 (x_1 + x_2)
    		  and  _parallel
    		= x_1 + x_2 -p^2 x_1 x_2
    		.

	In both cases, the number of master integrals (<ref>) is
	G
		=-χ(∖()) 
		= χ (∩())
		= χ(∩(1+x_1))
		= χ(-1)
		=1 according to (<ref>).

Much more on the Grothendieck ring calculus of graph hypersurfaces () can be found, for example, in <cit.> and <cit.>. These techniques can be used to prove some general statements about the counts of master integrals. Let us give just one example:


	Let G be a Feynman graph with a subgraph γ such that all propagators in γ are massless and γ has only two vertices which are connected to external legs or edges in G∖γ.[
		Such a graph γ is called massless propagator or p-integral.
	]
	Write G' for the graph obtained from G by replacing γ with a single edge (see figure <ref>), then
	
    G = γ·G'
    		.



	Every spanning tree T of G restricts on γ either to a spanning tree or to a spanning two-forest. In the first case, T∖γ is a spanning tree of G/γ (the graph where γ is contracted to a single vertex); in the second case, T∖γ is a spanning tree of G∖γ. Note that the two-forests T∩γ in the second case determine _γ from (<ref>), since all propagators in γ are massless. Therefore, we find
	_G = _γ·_G/γ
		     + _γ' ·_G∖γ
	where we set _γ'_γp^2=-1.
	Going through the same considerations for _G shows that
	
    _G 
    		     = _γ·_G/γ
    		     + _γ'·_G∖γ
    		.

	Now label the edges in γ as 1,…,N_γ and rescale all Schwinger parameters x_e with 2≤ e ≤ N_γ by x_1. Due to the homogeneity of _γ and _γ from corollary <ref>, we see that [^N ∖(_G)] = [^N ∖(A+x_1 B)] where A=U_γ_G/γ and B=_γ' _G∖γ in terms of U_γ_γx_1=1 and _γ' _γ'x_1=1.
	Applying (<ref>), we obtain a separation of variables:
	
    χ(^N ∖(_G))
    		   = - χ( ^N-1∖(_γ_γ' _G/γ_G∖γ) )
    		
       = - χ( ^N_γ-1∖(_γ_γ') )
    		·χ( ^N-N_γ∖(_G/γ_G∖γ) )
    		
       = - χ( ^N_γ∖(_γ) )
    		·χ( ^N-N_γ+1∖(_G') )
    		.

	In the last line we used (<ref>), upon noting the contraction-deletion formula _G' = _G/γ + x_0 _G∖γ in terms of the additional Schwinger parameter x_0 for the (massless) edge that replaces γ in G' (this formula is easily checked by considering which spanning trees and forests contain this edge or not).
	Note that for the subgraph γ, the value of p^2 does not matter for (_γ)=(_γ'), as long as p^2≠ 0. Finally, recall (<ref>).

Of course, this result is well-known on the function level: If G has a 1-scale subgraph γ, then the Feynman integral of G factorizes into the product

    _G(ν)
    	= _γ(ν_γ)p^2=-1·_G'(_γ,ν')

of the integrals of γ and G'. Here, we denote by ν_γ and ν' the indices corresponding to the edges in γ and outside γ, respectively, such that ν=(ν_γ,ν'). 
Note that the edge replacing γ in G' (see figure <ref>) gets the index _γ = ∑_e ∈γν_e - _γ·(/2) from (<ref>), which depends on the indices of γ (_γ denotes the loop number of γ).


	Let G be a graph with a pair e,f of massless edges in series or in parallel. Then G = G' where G' is the graph obtained by replacing the pair with a single edge.
	In other words, repeated application of the series-parallel operations from figure <ref> does not change the number of master integrals.


	Combine lemma <ref> with example <ref>.






§ TOOLS AND EXAMPLES




The Euler characteristic of a singular hypersurface can be computed algorithmically via several methods.[
	We will not discuss Kouchnirenko's theorem <ref> here, because in most examples we found that it does not apply. It seems that coefficients of graph polynomials are often not sufficiently generic.
]
In this section we demonstrate how some of these techniques can be used to compute the number of master integrals in various examples.

We begin with methods based on fibrations. In particular, the Euler characteristic can be computed very easily for the class of linearly reducible graphs, see section <ref>.
However, the decomposition of the Euler characteristic of the total space E of a fibration E⟶ B with fibre F into the product

    χ(E) = χ(B)·χ(F)

is true in general and not restricted to the linear case.
In section <ref> we use a quadratic fibration in order to count the master integrals of all sunrise graphs.

Apart from these geometric approaches, which seem to work very well for Feynman graphs, there are general algorithms for the computation of de Rham cohomology and the Euler characteristic of hypersurfaces. In section <ref> we discuss some available implementations of these algorithms in computer algebra systems.

In the final section <ref>, we comment on the relation of our result to other approaches in the physics literature.





 §.§ Linearly reducible graphs





If the polynomial (f)=a+x_N b is linear in a variable x_N, we saw in lemma <ref> that we can easily eliminate this variable x_N in the computation of the Euler characteristic (or the class in the Grothendieck ring) of the hypersurface (f) (or its complement).
Analogous formulas also exist in the case of a variety (f_1,…,f_n) of higher codimension, given that all of the defining polynomials f_i=a_i + x_N b_i are linear in x_N. Such linear reductions have been used heavily in the study of graph hypersurfaces, and are straightforward to implement on a computer <cit.>.

If such linear reductions can be applied repeatedly until all Schwinger parameters have been eliminated, the graph is called linearly reducible <cit.>. Linear reducibility is particularly common among graphs with massless propagators; we give some examples in figure <ref>.


At the end of a full linear reduction, the class of ^N∖() in the Grothendieck ring is expressed as a polynomial in the Lefschetz motive . To get the Euler characteristic, one then merely needs to substitute =[^1] ↦χ(^1)=1.

	Consider the 2-loop propagator 3' from figure <ref>. Linear reductions give
	
    [^5 ∖(_3')]
    		= -^4+5^3-13^2+21-15

	in the Grothendieck ring. Substituting ↦ 1 shows that 3'=3 via (<ref>).

In this way, we calculated the Euler characteristics for the graphs in figure <ref>, using an implementation of the linear reductions similar to the method of Stembridge <cit.>.
Our results are listed in table <ref>.




Beyond the computation of such results for individual graphs, it is possible to obtain results for some infinite families of linearly reducible graphs. In particular, efficient computations are possible for graphs of vertex width three <cit.>. For example, the class in the Grothendieck ring of () was computed for all wheel graphs in <cit.>.
It is possible to adapt such calculations to our setting (where the ambient space is ^N instead of ^N). For example, we could prove

	The number of master integrals of the massless propagators obtained by cutting a wheel  with  loops, either at a rim or a spoke (see figure <ref>), is

    '
    	= ”
    	= (-1)/2
    	.



The proof of this and related results will be presented elsewhere.






 §.§ Sunrise graphs



In <cit.>, the number of master integrals was computed for all sunrise integrals; based on a Mellin-Barnes representation and the differential reduction <cit.> of an explicit solution in terms of Lauricella hypergeometric functions.
To our knowledge, this has hitherto been the only non-trivial
[
	We consider families that arise simply by duplication of massless propagators, like those shown in <cit.>, as trivial (due to corollary <ref>).
]
infinite family of Feynman integrals with explicitly known master integral counts.
Their first result can be phrased as[
	Beware that the number 2^+1--2 given in <cit.> counts only irreducible master integrals, which means that it discards the +1 integrals associated to the subtopologies obtained by contracting any of the edges. Our conventions, however, do take these integrals into account.
]


	The -loop sunrise graph  from figure <ref> with +1 non-zero masses (and non-exceptional external momentum) has = 2^+1-1 master integrals.

We will now demonstrate that this result can be obtained from a straightforward computation of the Euler characteristic, according to corollary <ref>.

	The graph polynomials (<ref>) for the sunrise graph are

    = ∑_i=1^+1∏_j ≠ i x_j
    	= ( ∏_i=1^+1 x_i ) ( ∑_i=1^+11/x_i)
    	  and   = (-p)^2 ∏_i=1^+1 x_i + ∑_i=1^+1 x_i m_i^2
    	.

We note that for the first term in (<ref>), we find that ==0 imply ∏_i x_i = 0, which has no solutions in the torus—hence, this term contributes χ(^) = 0. We thus obtain

    (-1)^
    	= χ( ^∖( 1+∑_i=1^ x_i^-1) )
    	+ χ( ^∖ X^_p^2),

where we introduced the notation

    X^_p^2(-p^2+[m_+1^2+∑_i=1^ m_i^2 x_i] ·[1+∑_i=1^ x_i^-1] )
    	⊂^
    	.

The first Euler characteristic in (<ref>) is readily evaluated to (-1)^ by applying (<ref>) repeatedly (being on the torus, we may replace x_i^-1 by x_i), so we conclude that

    = 1
    	+ (-1)^·χ( ^∖ X^_p^2)
    	.

Now let us consider the projection π^⟶^-1 that forgets x_.
Set y  m_+1^2+∑_i< m_i^2 x_i and z  1+∑_i< x_i^-1, such that X_p^2^ = x_ p^2 = (1+z x_)(y+ m_^2 x_)⊂^. We note that the discriminant D of this quadric in x_ factorizes into

    D=(m_^2 - p^2 + yz)^2-4m_^2 yz
    	= ( yz - [p+m_]^2 ) ·( yz - [p-m_]^2 )
    	,

such that ^-1⊂(D) = X^-1_(p+m_)^2 X^-1_(p-m_)^2 is the disjoint union of two hypersurfaces.[
	We assume p^2 ≠ 0 and m_^2 ≠ 0, which guarantees that (p+m_)^2 ≠ (p-m_)^2.
]
Since the factors are related to the (-1)-loop sunrise by (<ref>), we find

    χ((D))
    	=2· (-1)^· (-1-1)
    	.

Over a point x'∈ X^-1_p± m_⊂(D) in the discriminant, the fibre of π^-1(x') has precisely one solution (x',x_) in X^L_p^2, determined by x_=-y/[m_ (m_± p)]:

    [(^∖ X^_p^2) ∩(D)]
    	= ([]-1)· [(D)]
    	.

If D(x')≠ 0 is non-zero and also yz ≠ 0, then the fibre π^-1(x') has precisely two distinct solutions x_L in the quadric X_p^2^. Hence, χ(π^-1(x'))=χ()-2=-2 and thus

    χ( (^∖ X_p^2^) ∖(yzD) )
    	= -2 χ( ^-1∖(yzD) )
    	= 2 χ((D)) + 2χ((yz))
    	,

where used that (D) ∩(yz)=∅ for non-exceptional values of p^2, such that (p ± m_)^2 ≠ 0 in (<ref>).
The reason that we need to exclude the case when yz=0 in (<ref>) is that for y=0, one of the solutions of X_p^2^ = x_p^2=(1+z x_)m_^2 x_ is x_=0 ∉; whereas for z=0 the equation for X_p^2^=x_p^2=y+m_^2 x_ becomes linear. 
In both cases, there is only one solution in the fibre, and there is none if both y=z=0 vanish (we assume p^2 ≠ m_^2):

    [(^∖ X_p^2^) ∩(yz)]
    	= ([]-2) · [(yz)] + [(y)] + [(z)]
    	.

We can now combine (<ref>)–(<ref>) via [Y]=[Y∩(D)]+[Y∩(yz)]+[Y∖(D· yz)] for Y=^∖ X_p^2^ into the reduction formula

    χ( ^∖ X^_p^2 )
    	= χ((D))
    	+ χ((y))
    	+ χ((z))
    	= 2·(-1)^·-1
    	.

Here, we inserted (<ref>) and used χ((y))=χ((z))=-χ(^-1∖(z))=(-1)^, which follows from repeated application of (<ref>)—just as above, when we computed the first term in (<ref>).
According to (<ref>), we can write the reduction as the recursion

    = 2-1 + 1,

which is obviously solved by the claimed =2^+1-1. It merely remains to verify the base case =1, and indeed, 1=1+χ(X_p^2^1)=3 follows easily from (<ref>) since X_p^2^1 = x_1 p^2 = (1+x_1)(m_2^2+m_1^2 x_1)⊂ consists of precisely two points.

It should be clear that our calculation can be adapted to the situation when some masses are zero. Let us demonstrate how to obtain another result of <cit.>:[
	The additional term -δ_0,-R in <cit.> subtracts a reducible integral that can be attributed to a subtopology. Our counting, however, accounts for all master integrals.
]

	The -loop sunrise graph with R ≤ non-zero masses, +1-R vanishing masses and non-exceptional external momentum, has L = 2^R master integrals.


	By corollary <ref>, we may replace all massless edges by a single (massless) edge without changing the number of master integrals; hence we can assume =R ≥ 1 (the totally massless case R=0 reduces to the trivial case of a single edge).
	Label the edges such that the massless edge is m_+1=0.

	We can apply the exact same recursion as in the proof of proposition <ref>; the only difference to (<ref>) is that now,
	χ((y)) = 0 vanishes because y=∑_i< m_i^2 x_i has become homogeneous in x such that [(y)]=[]·[(y)∩x_1=1]. Therefore, (<ref>) takes the form
χ(^∖ X_p^2^)
		= (-1)^·( 2-1-1 )
	and yields, via (<ref>), the recursion
	
    = 2 -1.

	We are done after verifying the base case: Indeed, 1=1+χ(X_p^2^1) = 2 from (<ref>) is clear since X_p^2^1 = p^2 x_1 = (1+x_1) m_1^2 x_1 is the single point x_1=p^2/m_1^2-1 in .







 §.§ General algorithms



The computer algebra system  <cit.> provides the function
http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html in the package http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/CharacteristicClasses/html/index.html. It implements the algorithm of <cit.> for the computation of the Euler characteristic.
This program requires projective varieties as input, so we need to homogenize  to = x_0  +, and can then use one of

    [^N∖(x_1 ⋯ x_N  )]
    	   =
    	[^N ∖ (x_1⋯ x_N )]
    	- [^N-1∖(x_1⋯ x_N )]
    	
       = 
    	[^N ∖ (x_0 x_1⋯ x_N )]

to express the sought after number of master integrals as the Euler characteristic of a projective hypersurface complement. We found that this algorithm performs well for small numbers of variables (edges): The examples in table <ref> require not more than a couple of minutes of runtime. For more variables, however, the computations tend to rapidly become much more time consuming and often impracticable.
Apart from the results in table <ref>, we also verified proposition <ref> for the sunrise graphs  using http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html for up to six loops.



	Consider the one-loop sunrise graph 1 with m_1^2=m_2^2=-p^2=1, which is a non-degenerate kinematic configuration. According to example <ref>, its Lee-Pomeransky polynomial is = (x_1+x_2)(x_1+x_2+1)+x_1 x_2. The  script
	
load "CharacteristicClasses.m2"
R=QQ[x0,x1,x2]
I=ideal(x0*x1*x2*((x1+x2)*x0+(x1+x2)^2+x1*x2))
Euler(I)
	
	computes the output 0 for χ((x_0 x_1 x_2 ) ∩^2). Using χ(^2)=3 and (<ref>), we conclude 1 = 3-0=3 in agreement with proposition <ref>.

Recall that the number of master integrals depends on the kinematical configuration; in table <ref> we give the results both for massless and for massive internal propagators. In particular, note how the massless 2-loop propagator 3' from example <ref> with only 3'=3 master integrals grows to carry 3'=30 master integrals in the fully massive case.

Furthermore,  also provides an implementation (the command http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.10/share/doc/Macaulay2/Dmodules/html/_de__Rham.html) of the algorithm <cit.> of Oaku and Takayama for the computation of the individual de Rham cohomology groups. This uses D-modules and Gröbner bases and tends to demand more resources than the method discussed above.

	Consider again the massive one-loop sunrise from example <ref>. The program
	
load "Dmodules.m2"
R=QQ[x1,x2]
f=x1*x2*(x1+x2+(x1+x2)^2+x1*x2)
deRham f
	
	computes the following cohomology groups of X=^2 ∖():
	H^0(X), H^1(X) ^3 and H^2(X)^5. Hence, 1=χ(X)=5-3+1=3 as in example <ref>.

The same functionality is provided by 's  library via the command
https://www.singular.uni-kl.de/Manual/4-0-3/sing_2398.htm. The  analogue of example <ref> is

LIB "deRham.lib";
ring R = 0,(x1,x2),dp;
list L = (x1*x2*(x1+x2+(x1+x2)^2+x1*x2));
deRhamCohomology(L);






 §.§ Comparison to other approaches



We successfully reproduced all of our results above (the wheels ' with ≤ 6, the sunrises  with ≤ 4 loops and the graphs from figure <ref> and table <ref>) with the program  <cit.>, which provides an implementation of Laporta's approach <cit.>. While it employs novel techniques to boost performance, in the end it solves linear systems of equations between integrals obtained from annihilators of the integrand of Baikov's representation (<ref>) in order to count the number of master integrals.

The observed agreement with our results is to be expected, since the identification of integral relations with parametric annihilators that we elaborated on in section <ref> works equally for the Baikov representation, which can also be interpreted as a Mellin transform.
Note, however, that we must use the options
 and 
for  in order to switch off the identification of integrals that differ by a permutation of the edges. The reason being that, in our approach, all edges e carry their own index ν_e and no relation between these indices for different edges is assumed.

Unfortunately, due to the way  treats subsectors, this can occasionally lead to an apparent mismatch. However, this is rather a technical nuisance than an actual disagreement.


	For the graph G in figure <ref>, the Euler characteristic gives G = 15, whereas both  <cit.> and  produce 16 master integrals. The problem arises from the subsector where the edges 1 and 2 are contracted: As shown in figure <ref>, it does have a remaining external momentum p_4, such that the momenta running through edges 3 and 4 are different—however, since p_4^2=0, the graph polynomials (and hence the Feynman integrals) are identical to those of the vacuum graph G' in figure <ref>. Since edges 3 and 4 in G' have the same mass, they can be combined and thus G' clearly has only a single master integral: the product of two tadpoles.

	But  and  instead consider the subsectors of G/1,2 obtained by contracting a further edge (3 or 4), and obtain the two tadpoles (see figure <ref>) consisting only of edges 4,5 and 3,5, respectively, as master integrals.
	Of course, these would be recognized as identical if symmetries were allowed; but the point is that even without using symmetries, there is only a single master integral for G' (as computed by the Euler characteristic).

Our results are also consistent with the conclusions obtained within the differential reduction approach <cit.>; indeed, we demonstrated in section <ref> how the master integral counts of <cit.> for the sunrise graphs emerge directly from the computation of the Euler characteristic.
Let us point out again, however, that some care is required for these comparisons, since those works refer to irreducible master integrals, which excludes integrals that can be expressed with gamma functions.
In particular, the fact that the two loop sunrise 2 with one massless line has 2=4 master integrals (see proposition <ref>) is consistent with <cit.>. We are counting all master integrals and are not concerned here with the much more subtle question addressed by the observation that two of these integrals may be expressed with gamma functions.

Finally, let us note that also the work of Lee and Pomeransky <cit.> addresses a different problem: Considering only integer indices ν∈^N, how many top-level master integrals are there for a graph G? This means that integrals obtained from subsectors (graphs G/e with at least one edge e contracted) are discarded. Geometrically, the number of the remaining master integrals is identified with the dimension of the cohomology group H^N(^N∖()).[
	Actually, they initially refer to a different, relative cohomology group; but in the description of their implementation in  they seem to work with this total cohomology group.
]
In most cases, the program  computes this number correctly, which then agrees with the other mentioned methods.[
	Occasional mismatches are known, like for the graph F_9 from figure <ref> that was addressed in <cit.>. These discrepancies are due to an error in the implementation of  that misses contributions from critical points at infinity (we thank Yang Zhang for bringing this to our attention).
]
We refer to <cit.> and <cit.> for detailed discussions of this comparison.
Note that the dimension (and a basis) of the top cohomology group can also be computed with the command
https://www.singular.uni-kl.de/Manual/4-0-3/sing_625.htm from the  library .

A direct comparison of  with our results is not possible, since the concept of top-level integrals does not make sense in our setting of arbitrary, non-integer indices ν. Here, there is no relation at all between integrals of a quotient graph G/e and integrals of G (the former do not depend on ν_e at all; the latter do).



	Using the inclusion-exclusion principle, one might be tempted to define

    G∑_γ⊆ G (-1)^γG/γ
    	= G - ∑_e G/e + ∑_e<fG/e,f - ⋯

	as the number of top-level master integrals, since it subtracts from all master integrals G the integrals associated to subsectors (and corrects for double counting). Note that if γ contains a loop, the corresponding term in the sum should be set to zero (we only consider contractions with the same loop number as G). The reverse relation,
	
    G
    		= ∑_γ⊆ GG/γ
    		,

	is consistent with the intuition that the total set of master integrals is obtained as the union of all top-level masters. 
	By
	_G/γ=x_e=0∀ e ∈γ,
	we find that
	
    G
    		= (-1)^Nχ( ^N∖() )

	is the Euler characteristic of the hypersurface complement inside affine space (as compared to the torus ^N as ambient space).
	However, this number can take negative values and may thus not be interpreted as a dimension. For example, G/1,2=-1 for the graph from figure <ref>, which is required for consistency of (<ref>) to get
	
    1
    		=G/1,2
    		=G/1,2 
    		+G/1,2,3 
    		+G/1,2,4 
    		= -1 + 1 + 1
    		.










§ OUTLOOK




We have studied linear relations between Feynman integrals that arise from parametric annihilators of the integrand ^s in the Lee-Pomeransky representation. Seen as a multivariate (twisted) Mellin transform, the integration bijects these special partial differential operators with relations of various shifts (in the indices) of a Feynman integral. In particular, every classical IBP relation (derived in momentum space) is of this type.

The question whether all shift relations of Feynman integrals (equivalently, all parametric annihilators of ^s) follow from momentum space relations remains open (see question <ref>). We showed that the well-known lowering and raising operators with respect to the dimension are consequences of the classical IBPs. A next step would be to clarify if the same applies to the relations implied by the trivial annihilators (<ref>).
Similarly, question <ref> asking whether the annihilator (^s)=^1(^s) is linearly generated, remains to be settled.
A positive answer to either of these would imply that the labor-intensive computation of the parametric annihilators could be simplified considerably ( is known explicitly, and ^1 can be calculated through syzygies).

The main insight of this article is a statement on the number of master integrals, which we define as the dimension of the vector space of the corresponding family of Feynman integrals over the field of rational functions in the dimension and the indices. Since we treat all indices ν_a as independent variables, this definition does not account for symmetries (automorphisms) of the underlying graph.
A natural next step is to incorporate these into our setup by studying the action of the corresponding permutation group.
The widely used partition of master integrals into top-level and subsector integrals, however, seems to be more difficult to take into account, as we touched on in remark <ref>.

Our result shows that the number of master integrals is not only finite, but identical to the Euler characteristic of the complement of the hyperspace =0 determined by the Lee-Pomeransky polynomial . This statement follows from a theorem of Loeser and Sabbah. We exemplified several methods to compute this number and found agreement with other established methods.
We expect that, combining the available tools for the computation of the Euler characteristic, it should be possible to compile a program for the efficient calculation of the number of master integrals for a wide range of Feynman graphs.

Let us conclude by emphasizing, once again, that the main objects of the approach elaborated here—the s-parametric annihilators generating the integral relations, and the Euler characteristic giving the number of master integrals—are well studied objects in the theory of D-modules and furthermore algorithms for their automated computation are available in principle.

In particular, we hope that this parametric, D-module theoretic and geometric approach can also shed light on the problems most relevant for perturbative calculations in QFT: the construction of a basis of master integrals, and the actual reduction of arbitrary integrals to such a basis. For this perspective, we would like to point out that our approach of treating the indices ν_a as free variables, in particular not tied to take integer values, is desirable in order to deal with dimensionally regulated integrals in position space, and for the ability to integrate out one-scale subgraphs (both situations introduce non-integer indices). For a recent step into this direction, see <cit.>.







§ INTEGRAL REPRESENTATIONS




In this appendix we add technical details on the material of section <ref>. We summarize the various well-known parametric representations, including their proofs, and the explicit relation to momentum space via propositions <ref> and <ref>. Furthermore, we give an alternative, algebraic proof for corollary <ref>. 






 §.§ Momentum space and Schwinger parameters






As in section <ref> we consider a connected Feynman graph G with N internal edges, +1 external legs and loop-number , which is related by Euler's formula =N-V+1 to the number V of vertices of G. Let us consider the case where each edge e of G is associated with a Feynman propagator,[We use the signature (1,-1,…,-1) for the Minkowski metric.]

    1/_e
    	= 1/-_e^2+m_e^2 -   
    	(1≤ e ≤ N),

which depends on the mass m_e of the particle e and the -dimensional momentum _e ∈^ flowing through this edge.
Enforcing momentum conservation at each vertex fixes all _e in terms of  independant external momenta _1,…,_ and  free loop momenta _1,…,_.
Note that the actual number of external legs of G is +1, since overall momentum conservation ∑_i=1^+1_i=0 imposes one relation among the external momenta. Taking only the inverse Feynman propagators _e as denominators, equation (<ref>) defines the Feynman integral associated to G.


	The graph in figure <ref> has V=5 vertices, N=6 internal edges and =2 loops. It depends on two independent external momenta p_1 and p_2. A choice of loop momenta and the resulting momentum flow is depicted in figure <ref>. With all masses zero, this yields
	
    (ν_1,…,ν_6)
    		= ∫_^[] _1/π^/2∫_^[] _2/π^/21/
    			  [-_1^2-]^ν_1
    			  [-_2^2-]^ν_2
    			  [-(_1-_2)^2-]^ν_3
    ×1/
    			  [-(_2+p_1)^2-]^ν_4
    			  [-(_2-p_2)^2-]^ν_5
    			  [-(_1-p_2)^2-]^ν_6
    		.


Typically, the number =(+1)/2+ of independent scalar products s_i,j in (<ref>) is larger than the number of edges in a graph G.
We can then extend the initial set of denominators (given as the inverse propagators of the graph) by a suitable choice of additional quadratic (or linear) forms in the loop momenta, such that we reach a set of  denominators with the property that the matrix  defined by

    _a
    	=∑_i,j∈_a^i,j s_i,j+λ_a

becomes invertible. This means that all loop-momentum dependent scalar products can be written as linear combinations of the denominators, see (<ref>). The additional denominators introduced in this way are called irreducible scalar products.

	We again consider the graph from figure <ref> with 6 internal edges labelled as in example <ref>. In the massless case, the inverse propagators are just _e = -_e^2 and their explicit decomposition into the =7 scalar products takes the form
	
    [ _1; _2; _3; _4; _5; _6;    ]
    
    
    
    
    
    
    
    
    		=
    		[      - _1^2;      - _2^2; - (_1-_2)^2; - (_2+_1)^2; - (_2-_2)^2; - (_1-_2)^2;             ]
    		=
    		[ -1  0  0  0  0  0  0;  0  0 -1  0  0  0  0; -1  2 -1  0  0  0  0;  0  0 -1  0  0 -2  0;  0  0 -1  0  0  0  2; -1  0  0  0  2  0  0;    ]_[  _1^2; _1 _2;  _2^2; _1 _1; _1 _2; _2 _1; _2 _2;       ]
    		+
    		[     0;     0;     0; -_1^2; -_2^2; -_2^2;       ]_λ
    		.

	The matrix  has rank 6 and annihilates (0,0,0,1,0,0,0)^. Thus we can choose _7 = _1 _1 as an irreducible scalar product to complete the basis of quadratic forms in the loop momenta. The matrix  then acquires an additional row (0,0,0,1,0,0,0) and becomes invertible.

Note that we assume that the inverse propagators of G (the initial set of denominators) are linearly independent (that is, the N × matrix  of the inverse propagators has full rank N) in order to be able to extend them to a basis of quadratic forms by choosing -N irreducible scalar products.
[
	If there are linear dependencies between the inverse propagators, these relations
	imply that the Feynman integral can be expressed in terms of contracted graphs with
	linearly independent inverse propagators.
	For example, if α_1+β_2 = 1, then iterated use of
		1/(_1 _2)=α/_2 + β/_1
	allows one to ultimately eliminate one of _1 or _2. Therefore, requesting
	linear independence is no restriction.
]

For each denominator _a we introduce a scalar x_a, which is known as Schwinger-, Feynman- or α-parameter. In definition <ref> we have introduced the decomposition 

    ∑_a=1^Nx_a_a
    	=-∑_i,j=1^_ij_i_j
    	+∑_i=1^ 2 _i_i
    	+
    	,

which determines a symmetric × matrix , a vector Q and a scalar . With their help we defined the polynomials ,  and =+ in (<ref>).
Explicitly, from definition <ref> and (<ref>) we can read off that

    _ij    = -1+δ_ij/2∑_a=1^Nx_a_a^i,j  for 1≤ i,j≤,
    _i    = 1/2∑_a=1^N∑_j=+1^Mx_a_a^i,j q_j
    	  for 1≤ i ≤ and
        = ∑_a=1^Nx_aλ_a
    	.

Since _ij is an × matrix with entries that are linear in the Schwinger parameters, the polynomial  is homogeneous of degree .
By Cramer's rule, ()^-1_ij is homogeneous of degree -1 and the linearity of  and  in the x_a implies


	 and  are homogeneous polynomials in the variables
	x_1,…, x_N with the degrees ()=
	and ()=+1. Hence, for =+, we have

    (∑_a=1^Nx_a∂_a)  
    	=  + (+1) 
    	= (+1)-
    	=  + .


Let us now come to the proof of proposition <ref> following <cit.> and <cit.>.

	We consider the Feynman integral defined in (<ref>),

    (ν_1,…,ν_N)
    	= (∏_j=1^∫[] _j/π^/2)
    	∏_i=1^N_i^-ν_i
    	.

	Using the Schwinger trick to exponentiate each denominator,[
		The integral (<ref>) converges only for (ν_a)>0 and therefore restricts the domain of convergence for the parametric integral. However, this has no consequences for algebraic relations, see remark <ref>.
	]
	
    1/_a^ν_a
    		= 1/Γ(ν_a)∫_0^∞ x_a^ν_a-1 e^-x_a _a x_a
    		,

	the integral in (<ref>) turns into

    (ν_1,…,ν_N)
    	=
    	(∏_i=1^N∫_0^∞x_i^ν_i-1 x_i/Γ(ν_i))
    	(∏_j=1^∫[] _j/π^/2)
    	e^-∑_a=1^Nx_a_a
    	.

	According to (<ref>) and (<ref>), we can complete the square in the exponent
	
    -∑_a=1^N x_a _a
    		= ( - ^-1)^ (-^-1) - /

	to perform the Gaußian integrals over the shifted loop momenta '  - ^-1 as[
		Recall that our metric has signature (1,-1,…,-1), so the integrations over the -1 spacelike components are Euclidean and give √(π^) each. The timelike integrations are understood as contour integrals and yield the same factor after rotating the integration contour to the imaginary axis, according to the Feynman -prescription.
	]
	
    (∏_j=1^∫[] '_j/π^/2) e^(')^'
    		= ()^-/2
    		= ^-/2
    		.

	In summary, we therefore arrive at the integral representation (<ref>):

    (ν_1,…,ν_N)
    	=
    	(∏_i=1^N∫_0^∞x_i^ν_i-1 x_i/Γ(ν_i))
    	e^-//^/2
    	.

We now multiply with 1=∫_0^∞δ(ρ-∑_j=1^N x_j) ρ and substitute x_a →ρ x_a.[
	Much more generally, we could replace ∑_j=1^N x_j in the δ-constraint with any other function as long as it is homogeneous of degree 1 and positive on _+^N.
]
The Jacobian ρ^N, the monomials x_i^ν_i-1 and δ(ρ-∑_j x_j) →δ(1-∑_j x_j)/ρ contribute the power ρ^ν-1, whereas the homogeneity of  and  from corollary <ref> implies that →ρ^ and /→ρ/. 
Overall, by realising that the integral over ρ is
	
    ∫_0^∞ρ^-1 e^-ρ/ρ
    		= Γ() ( /)^
    		,

we arrive at the first parametric formula (<ref>).
Similarly, we multiply the integrand of (<ref>) with 1=∫_0^∞δ(ρ-∑_i x_i) ρ and substitute x_i →ρ x_i. Using →ρ^ and →ρ^+1 from corollary <ref>, the integral over ρ becomes
	
    ∫_0^∞ρ^-1 ( + ρ)^-/2
    		= ^-/2( /)^Γ()Γ(/2-)/Γ(/2)

	and combines with the prefactors in (<ref>) to reproduce (<ref>).

We conclude the section with the proof of proposition <ref> following Grozin <cit.>:

	The action of ij on the integrand from (<ref>) is

    ijf 
    	= δ_ijf+f∑_a=1^N-ν_a/_aq_j∂_a/∂ q_i.

	According to (<ref>), the chain rule gives

    q_j∂/∂ q_i_a 
    	=
    	q_j∂/∂ q_i∑_k,m∈_a^k,m q_k q_m
    	= ∑_m=1^M_a^i,m(1+δ_i,m)q_jq_m

	and we can express the scalar products q_jq_m with j,m∈ in terms of denominators using (<ref>). The remaining terms with j,m> are products of external momenta, so

    ij f 
    	    = dδ_ijf-f∑_a,b=1^N C_aj^biν_a/_a(_b-λ_b)
    	  for   1≤ j≤  and
    ijf 
    	    = -f∑_a,b=1^N C_aj^biν_a/_a(_b-λ_b)-f∑_a=1^N∑_m=+1^M_a^i,mq_jq_mν_a/_a  if   <j ≤ M.

We conclude by noticing that multiplying the integrand f with ν_a/_a is equivalent to the action of the operator a defined in (<ref>), whereas multiplication with _b lowers the index ν_b and corresponds to b.







 §.§ Algebraic proof for corollary <ref>




With the proof of corollary <ref> we have shown that, for every momentum space IBP relation, there is a corresponding annihilator in ( ^-/2). The proof
rests on the inverse Mellin transform, which may be seen as a convenient but rather abstract argument. As a more direct alternative, we prove the statement in a purely algebraical way by use of properties of the graph polynomials.  
 
	
	The operators ij from (<ref>) and (<ref>) corresponding to the momentum space IBP relations annhilate the parametric integrand ^-/2:
	
    ij∈( ^-/2)
    		  for all 1≤ i ≤ and 1≤ j ≤ M.




	Let us first consider the case j≤. After acting with ij from (<ref>) on ^-/2 and dividing by (/2) ^-/2-1, we are left to prove the vanishing of
	
    2δ_i,j
    		-∑_a,b C^bi_aj x_a ( ∂_b  - λ_b [ +1-∑_c x_c ∂_c ] )
    		=2δ_i,j
    		- ∑_a,b C^bi_aj x_a ( ∂_b  - λ_b )

	where we exploited the homogeneity from (<ref>). Using (<ref>), we note that
	
    ∂_b  - λ_b 
    		= ∂_b [ (1+  + ^^-1) ] - λ_b 
    		= ∂_b / + ∂_b( ^^-1)

	because ∂_b  = λ_b according to (<ref>).
	In order to evaluate ∂_b with Jacobi's formula (∂_b )/ = ∑_r,s=1^^-1_r,s∂_b _s,r, we use (<ref>) to compute
	
    ∑_b ^b_m,j∂_b _s,r
    		= -1+δ_s,r/2∑_b ^b_m,j_b^s,r
    		= -1+δ_s,r/2δ_m,j,r,s
    		= -δ_m,rδ_j,s + δ_m,sδ_j,r/2

	which restricts m to either r or s. So in particular, m≤ and we can use (<ref>) in
	
    ∑_a,b C^bi_aj x_a ∂_b _r,s
    		=- ∑_a,m x_a _a^i,m1+δ_i,m/2( δ_m,rδ_j,s + δ_m,sδ_j,r) 
    		= _i,rδ_j,s + _i,sδ_j,r

	which proves that for arbitrary j (independent of whether j≤ or j>)
	
    ∑_a,b C^bi_aj x_a ∂_b /
    		= ∑_r,s=1^^-1_r,s
    		  (_i,rδ_j,s + _i,sδ_j,r)
    		= 2 δ_i,j
    		.

	Via (<ref>), this identity reduces the proof of (<ref>) to showing that
	
    ∑_a,b C^bi_aj x_a ∂_b ( ^^-1)
    		=
    		∑_a,b C^bi_aj x_a 
    		[
    			2 (∂_b )^^-1
    			- ^^-1 (∂_b ) ^-1]

	vanishes. The last term is easily evaluated with (<ref>) and gives
	
    ∑_a,b C^bi_aj x_a ^^-1 (∂_b ) ^-1
    		= 2∑_r,s=1^ (^-1)_r (^-1)_s _i,rδ_j,s
    		= 2 _i (^-1)_j
    		,

	whereas the derivative ∂_b can be read off from (<ref>) and the sum over b yields
	
    ∑_b _m,j^b (2∂_b _s)
    		= ∑_b _m,j^b ∑_r>_b^s,r q_r
    		= ∑_r> q_r δ_m,j,s,r
    		= ∑_r> q_r δ_m,rδ_s,j

	because j≤< r excludes the possibility that m=s and r=j. Thus with (<ref>),
	
    ∑_a,b C^bi_aj x_a (2∂_b )^^-1
    		=
    		(^-1)_j ∑_a ∑_m> x_a _a^i,m q_m
    		= 2 _i (^-1)_j

	cancels the contribution from (<ref>) in (<ref>) and finishes the proof in the case j≤.
	If instead we have j>, then we must replace δ_m,j,s,r = δ_j,rδ_m,s in (<ref>) such that
	
    ∑_a,b C^bi_aj x_a (2∂_b )^^-1
    		= ∑_s=1^∑_a _a^i,s (1+δ_i,s)
    		q_j (^-1)_s
    		= -2 ∑_s=1^_i,s (^-1)_s
    		= -2 Q_i q_j

	where we used (<ref>) once more.
	Now recall that (<ref>) remains true and becomes zero for j> because δ_i,j=0 since i≤. For the same reason, δ_j,s=0 in (<ref>) and therefore, using (<ref>),
	
    -∑_a,b C^bi_aj x_a (∂_b  - λ_b )
    		=
    		-∑_a,b C^bi_aj x_a (2∂_b )^^-1
    		= 2 _i q_j.

	This is precisely cancelled by the additional contribution to (<ref>) coming from ij in (<ref>) in the case j>: The additional term acts on ^-/2 as
	
    -∑_a ∑_m>_a^i,m q_j q_m x_a [  + 1-∑_c x_c ∂_c ] 
    		= - q_j ∑_a ∑_m>_a^i,m x_a q_m
    		= - 2 q_j _i

	after dividing by (/2) ^-/2-1. Note that here we used (<ref>) and (<ref>).





 §.§ The Baikov representation



In this section we discuss the representation of Feynman integrals suggested by Baikov in <cit.>, whose complete form (<ref>) was given by Lee in <cit.>. We will give some details on the derivation of this formula (see also <cit.>), which was presented in <cit.> and applied in our discussion of the lowering dimension shift in section <ref>.

Assume that q_1, …, q_ are vectors in a Euclidean vector space and write

    V_n [ q_n · q_n         ⋯  q_n · q_;         ⋮         ⋱         ⋮;   q_· q_n         ⋯    q_· q_;           ]
    	= ( q_i · q_j )_n ≤ i,j≤  and  
    	G_n  V_n

for their Gram matrices and determinants. Note that

    V_n = [     q_n^2  q_∙· q_n; q_n · q_∙     V_n+1;           ]  where  
    	q_∙· q_n
    	[ q_n+1· q_n;          ⋮;    q_· q_n;            ],
    	  
    	q_n · q_∙(q_∙· q_n)^

and thus, by adding -(p_n · p_∙) V_n+1^-1 times the lower -n rows to the first row,

    G_n/G_n+1 =
    	q_n^2-q_n+1,…,q_(q_n)^2
    	= q_n+1,…,q_^⊥(q_n)^2
    	.

Indeed, the formula q_n+1,…,q_ (v) = ∑_i,j=n+1^ q_i ( V^-1_n+1) _i,j (q_j · v) for the orthogonal projection of v onto the space spanned by q_n+1,…,q_ shows that

    q_n+1,…,q_(q_n)^2
    	   = ∑_i,j,k,l=n+1^(q_i · q_k)_(V_n+1)_i,k(V^-1_n+1)_i,j (q_j · q_n)
    		(V^-1_n+1)_k,l (q_k · q_n)
    	
       = ∑_j,k,l=n+1^δ_k,j (q_j · q_n)
    		(V^-1_n+1)_k,l (q_l · q_n)
    	= ∑_k,l=n+1^ 
    		(q_k · q_n)
    		(V^-1_n+1)_k,l (q_l · q_n)
    	
       = (q_n· q_∙)V^-1_n+1 (q_∙· q_n)
    	.

Now assume our integrand f only depends on the scalar products s_i,j = q_i · q_j, and we want to integrate out the first loop momentum q_1. Let us decompose q_1 = q_⊥ + q_∥ into the component q_∥∈q_2,…,q_ that lies in the space spanned by the other momenta, and the component q_⊥ in its orthogonal complement.
According to (<ref>), G_n^1/2 is the volume of the parallelotope spanned by q_n,…,q_.
Hence, changing coordinates from q_∥ to (s_1,2,…,s_1,) yields 

    ∫_ s_1,2 ⋯∫_ s_1,
    	= √(G_2)∫_^-1[-1] q_∥
    	.

The integral over the orthogonal component is, due to s_1,1=q_1^2=q_⊥^2 + q_∥^2, given by

    ∫_^-+1[-+1] q_⊥
    
    
    	   = π^(-+1)/2/Γ(-+1/2)∫_0^∞ q_⊥^2 (q_⊥^2 )^(--1)/2
    	= π^(-+1)/2/Γ(-+1/2)∫_q_∥^2^∞ s_1,1( G_1/G_2)^(--1)/2
    	.

Note that the lower boundary s_1,1=q_∥^2 corresponds to 0=q_⊥^2=G_1/G_2. Altogether,

    ∫_^[] q_1/π^/2 f(s)
    	=
    	π^(1-)/2/Γ(-+1/2)∫_G_1/G_2>0[] s_1,∙  f(s)
    	G_1^(--1)/2/G_2^(-)/2
    	.

Transforming the remaining loop integrations analogously, all but two of the Gram determinants cancel, and we conclude that

    ∏_i=1^∫[] _i/π^/2
    	f(s)
    	= π^- E/2-(-1)/4/Γ( --E+1/2) ⋯Γ( -E/2)∫[N] s_∙,∙ f(s)
    	·G_1^(--1)/2/G_+1^(-E-1)/2

where =+ is the sum of the number  of loops and the number  of linearly independent external momenta. Note that G_+1 =  (_i ·_j)_1≤ i,j≤ is the Gram determinant of the external momenta and independent of the integration variables.

	Since every denominator _e is a linear combination of s_i,j (and some loop momentum independent constant λ_e) according to (<ref>), an affine change of variables allows us to integrate over the values of _e's instead of s_i,j's. This transformation only introduces a constant Jacobian c' =  (^a_i,j). We set f(s)=∏_e _e^-ν_e in the formula (<ref>) above.
	The (Euclidean) integration domain is determined, according to (<ref>), by 
	0 < q_n+1,…,q_^⊥(q_n)^2 = G_n/G_n+1
	for 1≤ n ≤. Therefore, a point on the boundary of the integration domain is determined by G_n=0 for some 1≤ n ≤ L, which is equivalent to a linear dependence q_n ∈q_n+1,…,q_ and hence implies G_1=0.

	Note that we have to analytically continue (<ref>) from Euclidean to Minkowski space in order to obtain the Feynman integral (<ref>). As Wick rotation turns ∫[] _k/(π^/2) exactly into the measure ∫[] _k/π^/2 on the left-hand side of (<ref>), we only have to remember that, due to our mostly-minus signature (1,-1,⋯,-1) of the Minkowsi metric, the Euclidean scalar products on the right-hand side of (<ref>) receive a factor (-1). For example, the × determinant G_1 turns into (-1)^_1; similarly, G_+1 becomes (-1)^.
	Overall, analytic continuation gives an additional factor of
	
    (-1)^N·(-1)^(--1)/2/(-1)^(--1)/2
    		= (-1)^/2· (-1)^N+(-1)/2-(-1)/2
    		.

	We absorb the last factor, together with the Jacobian c', into the constant prefactor c, and have thus finally arrived at (<ref>).






§ THE THEORY OF LOESER-SABBAH




This section is devoted to the theorem <ref>, which was first stated in <cit.>. Beware that the original argument is flawed; a correct (but terse) proof was given in <cit.>. Our aim here is to provide a simplified and more detailed derivation.

Throughout we will consider modules  over the algebra N_k=N_k[x^-1] of differential operators (<ref>) on the torus in some number N of variables x_i, over some field k of characteristic zero. To lighten the notation, let us abbreviate θ_i  x_i ∂_i and set

    (θ_i,…,θ_N)
    	 k(θ_i,…,θ_N) _k[θ_i,…,θ_N]

for the algebraic Mellin transform <cit.>. We begin with the finite-dimensionality, which was proven in <cit.>:

	Let  denote a holonomic N_k-module. Then, for any 1≤ i ≤ N, its algebraic Mellin transform (θ_i,…,θ_N) is a holonomic i-1_k(θ_i,…,θ_N)-module.
	


	
	The full Mellin transform (θ_1,…,θ_N) is a finite-dimensional vector space over the field k(θ_1,…,θ_N).


	Since (θ_i,…,θ_N) = [(θ_i+1,…,θ_N)](θ_i), it suffices (by induction over i) to consider the case i=N.
	Introducing a new indeterminate ν, we extend the scalars from k to k[ν] to obtain a N_k[ν]-module [ν] _k k[ν]. It sits in an exact sequence
	
    0 ⟶[ν] [ν] 𝔐⟶ 0

	of N-1_k[ν]-modules, where 𝔐 = denotes the initial module  with the action of ν defined as ν m  -x_N ∂_N m.
	Since k(ν) is flat, this sequence remains exact after tensoring with k(ν) over k[ν]. Through identification of ν with -θ_N, we conclude that
	
    _k k(ν)/(∂_N + ν/x_N) (_k k(ν))𝔐_k[ν] k(ν)
    		_k[θ_N] k(θ_N)
    		= (θ_N)

	are isomorphic as N-1_k(ν)-modules. The left hand side is the quotient x_N^ν/∂_N  x_N^ν of the N_k(ν)-module x_N^ν_k k(ν) defined by the original action of N-1_k and x_N^± 1 on , but twisting the operator ∂_N to act like ∂_N+ ν/x_N.[
		This just encodes the natural action ∂_N x_N^ν m = x_N^ν(∂_N + ν/x_N) m on products of elements m of  with the function x_N^ν—hence the suggestive notation x_N^ν.
]
	The holonomicity of  implies that x_N^ν is also holonomic,[
		Given a good filtration Γ_∙ of , Γ_j' ( x_N^ν)  (x_N^-jΓ_2j) _k k(ν) defines a filtration of x_N^ν with _k(ν)Γ_j' ( x_N^ν) ≤_k Γ_2j≤ c · (2j)^N for some c<∞.
	]
	and hence its push-forward π_∗ ( x_N^ν) =  x_N^ν / ∂_N ( x_N^ν) (θ_N), with respect the projection π[k(ν)]^N [k(ν)]^N-1 that forgets the last coordinate, is also holonomic.

Now we want to relate this dimension to the de Rham complex
()
	=
	∂,
which is a special case of the Koszul complex:

	For commuting k-linear endomorphisms s=(s_1,…,s_N) of a k-vector space , let

    s_1,…,s_N( Λ^∙ k _k [N], )

	denote the Koszul complex with r-forms sitting in degree r-N. The r-N cochains

    [r-N]s 
    	= Λ^r 
    	= ⊕ _I=rI

have natural coordinates with respect to the basis Ii_1∧⋯∧i_r∈Λ^r k indexed by r-sets I=i_1<⋯<i_r. The cochain map reads (I m) = ∑_i ∉ I (i∧I)  s_i m.


Since θ_j (∏_i∈ I x_i )m = (∏_i ∈ I∪j x_i) ∂_j m (for j ∉ I), the rule

    ∂⟶θ,
    	  I m ↦I x^I m

defines a cochain map. It has an inverse, defined by I m ↦I x^-Im. We thus conclude that ()=∂ and θ are quasi-isomorphic and therefore share the same Euler characteristic.


We prove theorem <ref> by an induction over the number of variables. The base case is

	
	If  denotes a holonomic 1_k-module, then 
	
    _k(θ_1)(θ_1) 
    		= χ()
    		_k /∂_1  - _k (∂_1)
    		.



	We can pick a generator of  (by holonomicity,  is cyclic as a 1_k-module) and extend it to a (finite) basis of (θ_1) as a vector space over k(θ_1), due to corollary <ref>. Let ⊂ denote the k[θ_1]-module generated by such a basis, hence (θ_1)=(θ_1). By construction, = 1_k  = ∑_j∈ x_1^j is exhaustively filtered by the finitely generated k[θ_1]-modules _j ∑_i=-j^j x_1^i.[
		Due to θ_1 x_1^i = x_1^i (θ_1+i) ⊆ x_1^i⊆_j, indeed _j is a k[θ_1]-module.
	]

	Since (θ_1)=(θ_1) = _1(θ_1) is finitely generated, there is a non-zero polynomial b(θ_1)∈ k[θ_1] such that b(θ_1) _1 ⊆. Therefore, using (θ_1 -1)x_1 = x_1 θ_1,
	
    b(θ_1 ∓ j) x_1^± (j+1)
    		= x_1^± j b(θ_1) x_1^± 1⊆
    		x_1^± j b(θ_1) _1
    		⊆
    		x_1^± j⊆_j

	shows that the polynomials b_j+1(θ_1)  b(θ_1+j)b(θ_1-j) ∈ k[θ_1] have the property b_j+1(θ_1)_j+1⊆_j. Let Z=b^-1(0) denote the zeroes of b_1=b^2, then note that the zeroes of b_j+1 are (Z+j) ∪ (Z-j) and get pushed away from zero for increasing j. In particular, there exists some j_0 ∈ such that  b_j(0) ≠ 0 for all j > j_0. For each such value of j, we can find u_j,v_j ∈ k[θ_1] such that 1 = u_j(θ_1) b_j(θ_1) + v_j(θ_1) θ_1; then
	
    m = 1· m
    		= u_j(θ_1) b_j(θ_1) m + v_j(θ_1) θ_1 m
    		∈_j-1 + v_j(θ_1) θ_1 m

	holds for every m ∈_j. This proves (θ_1) ∩_j ⊆_j-1 for all j>j_0, and therefore (θ_1) ⊆_j_0. Similarly, we conclude / (θ_1 ) _j_0 / (_j_0∩θ_1). But given some m =θ_1 x ∈_j_0 with x ∈_j,
	x = u_j(θ_1) b_j(θ_1) x + v_j(θ_1) m
		∈_j-1 + _j_0
	proves that _j_0∩θ_1(_j) = _j_0∩θ_1(_j-1) for all j>j_0. In consequence, we have proven that
	
    (∂_1)
    		= 
    		 (θ_1) = (θ_1_j_0)
    		  and  /∂_1()/θ_1()_j_0/θ_1(_j_0)
    		;

	in other words, the Koszul complexes ()=∂_1 and _j_0θ_1 are quasi-isomorphic (see remark <ref>). The statement of the theorem thus reduces to the identity
	
    _k(θ_1)_j_0(θ_1) 
    		= χ(_j_0θ_1)

	for a finitely generated k[θ_1]-module _j_0. Since both sides are additive under short exact sequences, this claim reduces (via a finite free resolution) to the case of a free rank one k[θ_1]-module, i.e. k[θ_1], which is clear: k[θ_1](θ_1) = k(θ_1) is of dimension one over k(θ_1), while (θ_1) = 0 is trivial and k[θ_1]/(θ_1 k[θ_1]) = k is one-dimensional.

With this starting point, we can now prove theorem <ref> by induction. In fact, the higher dimensional case can be seen as a straightforward corollary of the univariate case above. In contrast to <cit.>, our demonstration avoids any reference to higher-dimensional lattices.

	Let  denote a holonomic N_k-module, and suppose we have proven theorem <ref> for all holonomic modules in less than N variables. In particular, we may invoke the claim for the N-1_k-modules
	∂_N and / ∂_N,
	as these are holonomic because they are the cohomologies of the complex
	
    0 ⟶⟶ 0

	which computes the push-forward of  along the projection π[k]^N ⟶[k]^N-1 that forgets the last coordinate. So we already know that 
	
    χ(/∂_N )
    		=_k'( /∂_N (θ') )
    		=_k''/∂_N '

	where k' k(θ') and ' (θ') with θ'(θ_1,…,θ_N-1). 
	Analogously, χ(∂_N) = _k'(∂_N'), where ∂_N' denotes the action of ∂_N on '. In conclusion, we know that
	
    χ( /(∂_N ) )
    		- χ( ∂_N )
    		   =
    		_k''/(∂_N ')
    		- _k'(∂_N')
    		= χ(')
    		
       = _k'(θ_N)'(θ_N)
    		=_k(θ)(θ),

	where we recognized the first line as the Euler characteristic of the de Rham complex of the 1_k'-module ' and applied theorem <ref> to get to the last line ('=(θ') is holonomic by lemma <ref>). So we only need to show that the left hand side is equal to χ().
	
	This is well-known and follows from the Grothendieck spectral sequence.[
		Let π^N[k]^N⟶pt denote the projection to a point, such that π^N = π^N-1∘π.
		The identity π^N_+ = π^N-1_+ ∘π_+ of the corresponding push-forwards in the derived category of N_k-modules implies that χ(())=χ(π^N_+()) = ∑_i (-1)^i χ( H^i (π_+ )), where H^-1(π_+ ) = ∂_N and H^0(π_+ ) = /(∂_N ).
	]
	Alternatively, an elementary way to obtain the identity χ(/(∂_N))-χ(∂_N)=χ() is given by the long exact sequence 
	
    ⋯→
    		H^i+1((∂_N)) →
    		H^i(()) →
    		H^i((/∂_N )) →
    		H^i+2((∂_N))
    		→⋯

	in de Rham cohomology <cit.>.

For completeness, let us demonstrate (<ref>), by following the standard construction in the proof of Kashiwara's theorem: First note that
	
    m∈∂_N^k m = 0   for some   k>0⊆

	defines a N_k-submodule of .[
		One only needs to check that x_N^± 1⊆, which follows from ∂_N^k+1 x_N^± 1 m = [(k+1)(∂_N x_N^± 1) + x_N^± 1∂_N) ∂_N^k m = 0 whenever ∂_N^k m =0.
	]
	The exact sequence
0 →→→/→ 0
	of holonomic N_k-modules induces a sequence of de Rham complexes,
0 →() →() →(/) → 0,
which is also exact (↦()=Λ^∙ k _k is exact by flatness of Λ^∙k).
	Hence we get a long exact sequence in cohomology:
	
    ⋯→
    		H^i(())→
    		H^i(()) →
    		H^i((/)) →
    		H^i+1(())
    		→⋯

	The key observation now is that ∂_N is injective on / and surjective on .[
		The first statement is clear since ∂_N ⊆. The second claim follows from the identity x^k∂^k = (∂ x -k)x^k-1∂^k-1 = ⋯ = ∏_i=1^k (∂ x -i), which implies 0=x_N^k ∂_N^k m = (-1)^k (k!) m ∂_N whenever ∂_N^k m = 0.
	]
	We thus get short exact sequences
	
    0 ⟶( ∂_N ) ⟶ 0
    		  and
       
    		0 ⟶///∂_N ⟶ 0

	of N-1_k-modules. The induced short exact sequences of de Rham complexes provide quasi-isomorphisms (∂_N) ()[1] and (/∂_N) (/), because the de Rham complex of a N_k-module  is the mapping cone of the map ∂_N∂'⟶∂'.
	Hence we obtain (<ref>) from (<ref>) due to
	
    H^i(())  H^i+1((∂_N))
    		  and  
    		H^i((/))  H^i((/(∂_N )))
    		.

To clarify this final step, first note that separating ∂_N from ∂' (∂_1,…,∂_N-1) yields an isomorphism of k-vector spaces

    Φ[∙-N]∂'⊕[∙-1-N]∂'   ⟶[∙-N]∂
    
    	= ⊕_ N ∉ II⊕⊕_N ∈ II
    
    
    
    
    	x ⊕ y
    	Λ^r-1∂'    ↦ 
    	x ⊕ (N∧ y)
    	.

In this representation, the differential is given by 

    Φ^-1( Φ(x ⊕ y) )
    	= '(x) ⊕ (∂_N x -'(y))
    	,

which is known as the mapping cone of ∂_N∂'⟶∂'. Here we denote by ' the differential of ∂'. 

If ∂_N is surjective, we can find x with ∂_N x = y and hence Φ(x⊕ 0)=Φ('x ⊕ y) for every y. Therefore, every element of ∂ has a representative of the form Φ(x ⊕ 0), modulo exact forms. But such a form is closed, Φ(x⊕ 0)=0, if and only if x ∈∂_N ∩'.

	If ∂_N is surjective, then ∂ and ∂_N∂'[1] are quasi-isomorphic.
	If ∂_N is injective, then ∂ and /∂_N ∂' are quasi-isomorphic.

The proof of the second statement is very similar to the surjective case and left as a straightforward exercise.




§ A TWO-LOOP EXAMPLE




We demonstrate some main points of this article by a pedagogical example. Consider the massless two-loop two-point graph with five propagators, graph 3' in figure <ref>. To this graph we associate the family of integrals

    (ν_1,…,ν_5)
    	= ∫[] l_1/π^/2∫[] l_2/π^/21/[-l_1^2]^ν_1 [-l_2^2]^ν_2 [-(l_2-p)^2]^ν_3 [-(l_1-p)^2]^ν_4 [-(l_1-l_2)^2]^ν_5

with two loop-momenta q_1=l_1, q_2=l_2 and one external momentum q_3=p. We normalize to -p^2=1. The graph polynomial =+ is given by the Symanzik polynomials 

    = (x_1+x_4)(x_2+x_3)+x_5(x_1+x_2+x_3+x_4)   and
        = x_1 x_2(x_3+x_4)+x_3x_4(x_1+x_2)+x_5(x_1+x_2)(x_3+x_4).

By definition <ref> the modified Feynman integral

    (ν_1,…,ν_5)
    	=  ^s

is related to the Feynman integral by 

    (ν_1,…,ν_5)
    	=Γ(-s)/Γ(-s-ω) (ν_1,…,ν_5)

with ω = ν_1+ν_2+ν_3+ν_4+ν_5+2s and s=-/2.




 §.§ From annihilators to integral relations


A set of generators of the annihilator ideal _A_5[s](^s) can be derived in  <cit.> with algorithms introduced in <cit.>. Using the command  we obtain a set of 13 generators. To give an impression, the first five of them read


    P_1
    	    = ∂_1-∂_2+(x_3+x_4+1)(∂_3-∂_4)+(x_3-x_4)∂_5,
    
    
    	P_2
            = (x_2-1)∂_1+(-2x_1-3x_2-1)∂_2+(-x_4+1)∂_3+(2x_3+3x_4+1)∂_4 +(2x_1-x_2 
        -2x_3+x_4)∂_5,
    
    
    P_3
        = (x_1+2)∂_1+(x_1+2 x_2)∂_2+x_4 ∂_3 +(-2 x_3-3 x_4-2)∂_4+(-x_1+2 x_3-x_4)∂_5,
    
    
    P_4
        = ∂_1 x_1+(x_2+1)∂_2+(-x_3-x_4-1)∂_3+(x_4+x_5)∂_5-s,
    
    
    P_5
        = (x_1 x_3+x_1 x_4+x_2 x_3+x_2 x_4+x_1+x_2+x_3+x_4)∂_4 +(-x_1 x_2-x_1 x_3-x_1 x_5
        -x_2 x_3-x_2 x_5-x_2-x_3-x_5)∂_5.

We notice that this set includes one generator which is quadratic in the differential operators, reading

    P_13    = (x_4+x_5-2)∂_1^2+(2 x_2+x_5)∂_2^2 +(-2 x_3^2+2 x_3 x_4-x_3 x_5+x_4 x_5)∂_5^2-x_3 x_4∂_3∂_5
       +(2 x_3^2+x_3 x_4-2 x_4^2+2 x_3-2 x_4)∂_4∂_5+(x_3-x_4+(-2 x_2-x_4-2 x_5+2)∂_2
       +(x_4^2-x_4)∂_3+(-x_4^2+2 x_3+3 x_4+2)∂_4+(x_3 x_4+x_3 x_5-x_4^2-x_4 x_5-4 x_3+2 x_4
       -x_5)∂_5)∂_1+(x_3 x_4∂_3-x_3+x_4+(-x_3 x_4-2 x_3-2 x_4-2)∂_4+(-2 x_2 x_3+2 x_2 x_4
       -x_3 x_5+x_4 x_5+2 x_3+x_5)∂_5+2)∂_2+(-x_3+x_4)∂_5.


Every operator in _A_5[s](^s) gives rise to an integral relation. According to lemma <ref>, we just need to replace each x_i by i and each ∂_i by -i to obtain a shift relation between modified Feynman integrals. For example, for the generator P_1, we obtain the shift operator

    P_1
    	=-1+2-(3+4+1)(3-4)-(3-4)5

satisfying 

    P_1(ν_1,…,ν_5)=0.

For the (unmodified) Feynman integral (ν_1,…,ν_5) we obtain the corresponding shift relation via (<ref>). From P_1 we obtain the operator 

    O_1
    	= (∑_i=1^5i+3s)(1-2+3-4)-(3+4)(3-4)-(3-4)5

which satisfies

    O_1(ν_1,…,ν_5)=0.






 §.§ Linear annihilators



It is useful to consider the linear annihilators ^1_N[s](^s) ⊆_N[s](^s). Recall that they are the annihilators which are linear differential operators, being of the form P=q + ∑_i=1^N p_i ∂_i with p,q_1,…,q_N ∈[s,x_1,…,x_N]. We compute the generators of ^1(^s) as generators of the Syzygy-module of (,∂_1 ,…,∂_N ), using the  command . For our example we obtain the following 8 generators:

    L_1
       = ∂_1-∂_2+(x_3+x_4+1)∂_3+(-x_3-x_4-1)∂_4+(x_3-x_4)∂_5,
    
    L_2
       = (-x_2-1)∂_1+(x_2+1)∂_2+(-x_4-1)∂_3+(x_4+1)∂_4+(x_2+x_4+2 x_5)∂_5-2s,
    
    L_3
       = (2 x_1+x_2+1)∂_1+(x_2+1)∂_2+(-2 x_3-x_4-1)∂_3+(-x_4-1)∂_4+(-x_2+x_4)∂_5,
    
    L_4
       = (x_1+x_2+1)∂_1+(-x_1-x_2-1)∂_2+∂_3-∂_4+(x_1-x_2)∂_5,
    
    L_5
       = -2s x_4+(x_4+x_5)∂_1+(-2 x_2-x_4-x_5-2)∂_2+(2 x_3 x_4+x_4^2+2 x_3+3 x_4+2)∂_3
        +(x_4^2+x_4)∂_4+(-x_4^2-2 x_4-x_5)∂_5,
    
    L_6
       = (-x_2 x_3+x_2 x_4-x_2-x_3+2 x_4+2 x_5)∂_1+(-2 x_1 x_3+2 x_1 x_4-x_2 x_3+x_2 x_4+2 x_1
        -9 x_2-x_3-2 x_5-6)∂_2+(-2 x_1 x_3-2 x_1 x_4-x_2 x_3-x_2 x_4-2 x_3^2-4 x_3 x_5+2 x_4^2
        -4 x_4 x_5-2 x_1-x_2+3 x_3+6 x_4-4 x_5+6)∂_3+(2 x_1 x_3+2 x_1 x_4+x_2 x_3+x_2 x_4
        +4 x_3 x_4+4 x_3 x_5+4 x_4^2+4 x_4 x_5+2 x_1+x_2+x_3+6 x_4+4 x_5)∂_4+(-2 x_4^2-6 x_4
        -4 x_5)∂_5+s(-4 x_4+2),
    
    L_7
       = (2 x_1 x_2+2 x_2^2+2 x_1+4 x_2+2)∂_1+(x_2 x_4+2 x_2+x_5)∂_3+(-x_2 x_4-2 x_2-2 x_4
        -x_5-2)∂_4+(-2 x_2^2-x_2 x_4-2 x_2 x_5-2 x_2-x_5)∂_5,
    
    L_8
       = (x_4+x_5)∂_1+(2 x_1 x_2+2 x_1 x_4+2 x_1 x_5+2 x_2^2+2 x_2 x_4+2 x_2 x_5+2 x_1+2 x_2+x_4
        +x_5)∂_2+(-2 x_1 x_3-2 x_1 x_4-2 x_3 x_5+x_4^2-2 x_1+2 x_3-x_4)∂_3+(-x_4^2-2 x_4 x_5-x_4
        -2 x_5)∂_4+(-x_4^2-2 x_4 x_5-2 x_5^2+x_5)∂_5+s(-2 x_2+2 x_5-2),

Using  we find that for our two-loop example every generator P_i can be expressed as a linear combination of the linear generators L_i over _5[s](^s). For instance, the first five annihilators satisfy

    P_1    = L_1,   
    	P_2 = -2L_1-L_3+2L_4,   
    	P_3 = 2L_1+L_3-L_4,   
    	2P_4 = L_2+L_3, 
    	
    
    	2P_5    = -2(x_1+x_2+1)L_1+(1+x_2-x_4-x_5)(L_2-L_3)-2(x_2+x_4+x_5)L_4+L_5+2L_7-L_8.

We emphasize that such a relation exists for all the P_i. In particular such a relation also exists for the quadratic P_13, which however is too long to be shown here. As a consequence we can view _N[s](^s) as generated by the linear L_i, which will simplify the discussion in section <ref>.




 §.§ From IBP relations to annihilators

Going in the other direction, we can derive annihilators from momentum-space IBP relations. In the usual way, inserting the differential operators 

    ij
    	= ∂/∂ q_iq_j
    	  for   i∈1,2  and   j∈1,2,3

we obtain six IBP relations ij=0 with the shift operators 

    11   =-41-51+52+4-21-4-5,
    	
    12   =-12+15-41-43+45-51+52+4-1+5,
    	
    13   =-s1+s4-41-51+52-53+14+54-1+4,
    	
    21   =-21+25-32-34+35+51-52+3-2+5,
    	
    22   =-32+51-52+3-2s-22-3-5
    23   =23-32+51-52+53-54-2-3-2+3
    	.


Following the steps in the proof of corollary <ref> we derive for each shift opeartor ij a parametric annihilator ij. We obtain

    11   = (x_1 x_4+2 x_1+x_4+x_5)∂_1+(x_2 x_4-x_5)∂_2+x_3 x_4∂_3-3s x_4+(x_4^2+x_4)∂_4+(x_4 x_5
        +x_5)∂_5-2s,
    12   = -3s x_4+(x_1 x_4+x_1+x_4+x_5)∂_1+(x_2 x_4+x_1-x_5)∂_2+(x_3 x_4+x_4)∂_3+x_4^2∂_4
        +(x_4 x_5-x_1-x_4-x_5)∂_5,
    13   = 3s x_1-3s x_4+(-x_1^2+x_1 x_4+x_1+x_4+x_5)∂_1+(-x_1 x_2+x_2 x_4-x_5)∂_2+(-x_1 x_3
        +x_3 x_4+x_5)∂_3+(-x_1 x_4+x_4^2-x_1-x_4-x_5)∂_4+(-x_1 x_5+x_4 x_5)∂_5,
    21   = -3s x_3+(x_1 x_3+x_2-x_5)∂_1+(x_2 x_3+x_2+x_3+x_5)∂_2+x_3^2∂_3+(x_3 x_4+x_3)∂_4
        +(x_3 x_5-x_2-x_3-x_5)∂_5,
    22   = -3s x_3+(x_1 x_3-x_5)∂_1+(x_2 x_3+2 x_2+x_3+x_5)∂_2+(x_3^2+x_3)∂_3+x_3 x_4∂_4+(x_3 x_5
        +x_5)∂_5-2s,
    23   = 3s x_2-3s x_3+(-x_1 x_2+x_1 x_3-x_5)∂_1+(-x_2^2+x_2 x_3+x_2+x_3+x_5)∂_2+(-x_2 x_3
        +x_3^2-x_2-x_3-x_5)∂_3+(-x_2 x_4+x_3 x_4+x_5)∂_4+(-x_2 x_5+x_3 x_5)∂_5.

These operators are useful to compare both approaches as discussed next. 



 §.§ Comparing annihilators and IBP operators
 

According to corollary <ref>, every momentum-space IBP relation corresponds to a parametric annihilator. For our two-loop example, this is given by the fact that 

    ij∈_5[s](^s)
    	  for  
    	i∈1,2  and  
    	j∈1,2,3.


We may furthermore ask if the reverse is true: Can every annihilator of  be derived from IBP relations? If the answer would be no, the approach via parametric annihilators would provide new integral identities. While this question remains open for the general case, we can test it for simple Feynman graphs such as the present two-loop example.

In a first attempt, we could consider the shift relations obtained from the generators P_1,…,P_13 and try to confirm that they are combined IBP relations. If we use one of the well-known implementations of Laporta's algorithm to reproduce e.g. equation (<ref>), we have to fix the values of ν_1,…,ν_5 and do not answer the question for arbitrary values of the ν_i. We therefore approach the problem on the level of parametric differential operators instead. 

We find that actually not all parametric annihilators are contained in ; however, they turn out to still be consequences of the momentum space IBP relations in the following sense:
While we checked that P_1 ∉, we can find a polynomial q_1∈[s,x_1∂_1,…,x_5∂_5] such that q_1 P_1∈. Recall that, under the Mellin transform, such a q_1 corresponds to a polynomial in the dimension and in the ν_e. 
The interesting question then is if we can find a polynomial q∈[s,x_1∂_1,…,x_N∂_N] for every P∈_N[s](^s) such that q P ∈. If we can find such a q_i for every generator P_i, we can express every annihilator in terms of the ij. The q_i are the denominators of the coefficients in such a linear combination.

In section <ref> we have seen for our example that _N[s](^s) is generated by the linear annihilators L_i. As a consequence, it is sufficient to show that for each L_i there is a q̃_i such that q̃_i L_i ∈ for i=1,…,8. Indeed we can construct such q_1,…,q_8 by an explicit Ansatz. For example we obtain the identity

    (2∑_i=1^5x_i ∂_i -6s)L_1= c_1 11 +c_2 21+c_3 12 + c_4 22 + c_5 13 + c_6 23

with

    c_1 
       = -(-2 ∂_1+ ∂_2- ∂_3+2 ∂_4+x_5 ( ∂_1 (s+1)+ ∂_2 (-s-1)+ ∂_5s) +x_4 ∂_4+x_3 ( ∂_2 (-s-1)
        - ∂_4s
    + ∂_5s+3s^2+3s+x_5 ∂_5 (-s-1)+x_4 ∂_4 (-s-1))+x_3^2 ∂_3 (-s-1)+x_2 (- ∂_1s
       + ∂_2 (
    -s-1)+ ∂_5s+x_3 ∂_2 (-s-1))+x_1 ( ∂_1 +x_3 ∂_1 (-s-1))),
    
    c_2
       = -(- ∂_2+ ∂_3-2s^2-2s+
    x_5 ∂_5 (2s+1)+x_4 (- ∂_3s+ ∂_4 (s+1) + ∂_5s)+x_3 ∂_3+x_2 ∂_2
       +x_1 ( ∂_1 (s+1)- ∂_2s+
     ∂_5s)),
    
    c_3
        =-( ∂_1+1- ∂_4+s +x_5 ( ∂_1 (-s-1)+ ∂_2 (s+1)- ∂_5s)-x_4 ∂_4+x_3 ( ∂_2 (s+
    1)+ ∂_4s
       - ∂_5s-3s^2-3s+x_5 ∂_5 (s+1)+x_4 ∂_4 (s+1))+x_3^2 ∂_3 (s+1)+x_2 ( ∂_1s+
     ∂_2 (s+1)
       - ∂_5s+x_3 ∂_2 (s+1))+x_1 (- ∂_1+x_3 ∂_1 (s+1)))
    
    c_4 
       = -(- ∂_1+2 ∂_2-2 ∂_3
    + ∂_4+x_5 ∂_5+x_4 (- ∂_3+ ∂_5)-x_3 ∂_3-x_2 ∂_2+x_1 (- ∂_2+ ∂_5))
    
    c_5
       =-( ∂_1- ∂_2+ ∂_5)
    
    c_6
       =
    -( ∂_1- ∂_2- ∂_5).

Using these results and the expressions for the generators of _5[s](^s) in terms of the L_i, we can derive every annihilator from the IBP operators. We have done this computation with the same conclusion for several further graphs of low loop-order. These computations support our conjectures phrased in questions <ref> and <ref>.



 §.§ The number of master integrals


The result of the previous subsection implies that the parametric approach and momentum-space IBP lead to the same number of master integrals for this example. Indeed, as mentioned in section <ref>, we compute the Euler characteristic

    = 3

and obtain the same number of master integrals with . The three master integrals suggested by  are

    I_1=(1,1,1,1,0),  
    	I_2=(0,1,0,1,1)   and   
    	I_3=(1,0,1,0,1).

Notice that symmetries of the graph were not taken into account here, which in  is assured by setting  and . For an integral reduction in practice, one would of course make use of the symmetry  

    (ν_1,ν_2,ν_3,ν_4,ν_5)=(ν_2,ν_1,ν_4,ν_3,ν_5)

and compute with one of the sets I_1,I_2, I_1,I_3.



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