Moment Varieties of Measures on Polytopes
Kathlén Kohn Boris Shapiro Bernd Sturmfels


Seeded Graph Matching via Large Neighborhood Statistics
    
Elchanan Mossel 

MIT

elmos@mit.edu
Jiaming Xu

Duke University

jiaming.xu868@duke.edu

    26.7.2018
==============================================================================================



 
The uniform probability measure on a convex polytope
induces piecewise polynomial densities on its projections. For a fixed combinatorial type of simplicial polytopes,
the moments of these measures are rational functions in the vertex coordinates.
We study projective varieties that are parametrized by finite collections of
such rational functions. Our focus lies on determining the prime ideals
of these moment varieties. Special cases include  Hankel 
determinantal ideals for polytopal splines on line segments,
and the relations among multisymmetric functions given by
the cumulants of a simplex. In general, our moment varieties
are more complicated than in these two special cases. They offer 
challenges for both numerical and symbolic computing
in algebraic geometry.
  



§ INTRODUCTION


Inverse moment problems for positive and real-valued measures have been an active area of research

since the 19th century when Stieltjes obtained
 first significant results in the one-dimensional case. One point of entry
 to this subject area is Schmüdgen's textbook <cit.>.

In applications one usually considers a restricted class of measures, e.g. those with finite  or
low-dimensional support, Gaussian mixtures, unimodal measures, just to mention a few. The set of moments  is often restricted as well, e.g. by degree or structure.
One classical situation occurs in logarithmic potential theory 
where one studies harmonic moments <cit.>.
 Such restrictions reveal many interesting features,
 such as the non-uniqueness of a measure with given moments (cf. <cit.>).
 Another feature that is important, but much less studied, is the
   overdeterminacy of the moment problem. This arises from 
   relations among the moments.
   
We are interested in polynomial relations among moments 
of probability measures on ^d. Such relations
exist for many natural families of measures <cit.>. They define
the moment varieties of these families.
For finitely supported measures these are  the
 secant varieties of Veronese varieties <cit.>.
Moment varieties of Gaussians and their mixtures were 
characterized in <cit.>.

In this paper we study moment varieties arising from realization spaces of
convex polytopes <cit.>. 
If P is a polytope in ^d, then  we
 write μ_P for the uniform probability distribution on P.
The moments of the distribution μ_P are the expected values of the monomials:

    m_i_1 i_2 ⋯ i_d   =   ∫_^d w_1^i_1 w_2^i_2⋯ w_d^i_d  dμ_P 
         for   i_1,i_2,…, i_d ∈.

 The list of all moments  (  m_I  :  I ∈^d )
 uniquely encodes  the polytope P since any positive or real-valued measure with compact support
  is determined by its full list of moments. 
  
The inverse moment problem for polygons and polytopes is still largely unexplored.
It has appeared in logarithmic potential theory <cit.>,
 and in connection with the mother body problem  <cit.>. 
  Algorithms for  reconstructing P from  its axial moments
  can be found in  <cit.>.
  A practical application for moments of planar polygons was suggested by
  Sharon and Mumford <cit.> 
  as a tool to reconstruct arbitrary planar shapes from their fingerprints.
  
 To introduce our topic of investigation,  suppose that P is a simplicial polytope 
 in ^d with n vertices, denoted x_k = (x_k 1,x_k 2,…,x_k d)
 for k=1,2,…,n.
One can vary these vertices locally without changing the
 combinatorial type  of the polytope P.
Following  <cit.>, by the combinatorial type
  we mean the lattice of faces of P.

For  fixed ,
 each moment
m_I, for I=(i_1,… , i_d), becomes a locally defined function of the n × d-matrix X = (x_k l).
We shall see in Section <ref> that this function is rational and therefore extends to a
dense set of matrices X.
Furthermore, it is homogeneous of degree |I|, i.e. m_I(t X) = t^|I| m_I(X).



[d=1, n=2]
 The polytope is a segment P = [a,b] on the real line  ^1.
Here a = x_11 and b=x_21.
The ith moment of the uniform distribution on P is found by calculus:

    m_i    =   1/b-a∫_a^b w^i   dw    =   1/i+1b^i+1 - a^i+1/b - a   =   1/i+1 (a^i + a^i-1 b + a^i-2 b^2 + ⋯ + b^i).

These expressions are the coefficients of the
normalized moment generating function

    ∑_i=0^∞  (i+1) · m_i  · t^i    =   1/(1 - a t)(1-b t) .

The parametrization (a,b) ↦  (m_0:m_1:⋯:m_r)
defines a surface in projective r-space ^r, for any r ≥ 3.
The first such moment surface, shown in Figure <ref>,
is defined by the equation

    2 m_1^3 - 3 m_0 m_1 m_2 + m_0^2 m_3   =    0.

This cubic surface in ^3 is
 singular along the line { m_0 = m_1 = 0} in the plane at infinity.
 It also contains the twisted cubic curve
{m_0m_2 = m_1^2,  m_0 m_3 = m_1m_2,    m_1 m_3 = m_2^2 }.
Points on that curve correspond to
degenerate line segments [a,a] of length zero. 


The objects studied in this paper generalize Example <ref>.
We fix a combinatorial type 𝒫 of simplicial d-polytopes
and a subset
𝒜⊂^d with 0 ∉𝒜.
Consider the semialgebraic set  of n × d matrices X whose rows are the
vertices of a polytope of  type .
This set is open in ^n × d. Each moment  m_I depends
rationally on X, so it extends to a unique rational function on ^n × d.
The vector of moments ( m_I  :  I ∈𝒜∪{0} )
 defines a rational map ^n × d^|𝒜|.
The moment variety  ℳ_𝒜() is the
closure of the image of this map.
By construction, ℳ_𝒜() is an irreducible
projective variety. Its dimension is nd, provided 𝒜 is big enough.
Our aim is to compute these moment varieties as explicitly as we can.
Of particular interest is the variety given by all moments of order ≤ r.
This is denoted

    ℳ_[r]()   ⊂  ^d+rd-1 .


If  lies in a coordinate subspace then 
we can reduce the dimensionality of our problem, 
but at the cost of passing to non-uniform measures on polytopes. 
 Suppose that ⊂^d^' for d^'<d
 and let π be the projection ^d →^d'.
 All moments  m_I with I ∈ of the original polytope P⊂^d are moments
  of the induced distribution    π_*(μ_P)   on  P^'=π(P) in ^d'.
  Its density at p ∈ P' is the (d-d^')-dimensional volume of the inverse image π^-1(p)⊂ P. In other words, π_*(μ_P)      is the push-forward of μ_P under the projection π.  Densities of such measures are piecewise polynomial functions of degree d-d^' and are called polytopal splines. They have been studied since the pioneering paper <cit.>;  for more details consult <cit.>. 

This paper is organized as follows. 
In Section <ref> we derive the parametric representation for our moment varieties.
It is encoded in a rational generating function 
(Theorem <ref>) whose numerator polynomial (<ref>)
is Warren's adjoint from geometric modeling <cit.>.
Section <ref> concerns the univariate distribution obtained
by projecting P onto a line segment P'. This corresponds to the above polytopal 
splines  with d'=1. Their moment varieties are
determinantal. 
Explicit Gröbner bases are furnished by the Hankel matrices in Theorem <ref>.

In Section <ref> we examine the case  when 𝒫 is the d-simplex.
We study moments and cumulants for 
uniform probability distributions on simplices, and we express
these as multisymmetric functions. This connects us to
an interesting, but notoriously difficult, subject in algebraic combinatorics.
Brill's equations  <cit.> are used to characterize
the moment varieties of simplices. 
A Grassmannian makes a surprise appearance in
Proposition <ref>.
The section concludes with tetrahedra: their moments of order ≤ 3
form a 12-dimensional variety in ^19.


Moment varieties of polytopes are invariant under affine transformations but
not under projective transformations. Section <ref> explores these
group actions and their invariant theory.
This is subtle because the affine group is not reductive, but
Theorem <ref> offers a remedy.

Section <ref> presents a computer-aided case study for quadrilaterals.
We calculate their moment hypersurfaces in ^9.  This includes
the invariant hypersurface of degree 18 in Theorem <ref>.
We also examine concrete issues of identifiability and symmetry,
seen in the fiber of ten quadrilaterals in Figure <ref>, and in
relations among moments of tetrahedra in Proposition <ref>.
Some of our results are proved by certified numerical computations  as in <cit.>.


Section <ref> offers a summary of this paper and an outlook for future directions.
Readers will find numerous open questions that arise from our
investigations in the earlier sections.








§ GENERATING FUNCTIONS



The moments of a polytope P can be encoded in a rational generating function.
We begin by explaining this for the special case n=d+1, when 
 P is a d-dimensional simplex Δ_d in ^d.
 The vertices of the simplex Δ_d are denoted by x_k = (x_k1,…,x_kd)
for k=1,2,…, d+1.



The moments m_I of the uniform probability distribution on
the simplex Δ_d  are obtained from the coefficients of
the normalized moment generating function

    ∏_k=1^d+11/1 - (x_k1 t_1 + x_k2 t_2 +⋯+ x_kd t_d)    =  ∑_i_1,i_2,…,i_d ∈(i_1 + i_2 +⋯+ i_d +d)!/i_1 !   i_2 ! ⋯  i_d!  d !
    m_i_1 i_2 ⋯ i_d t_1^i_1 t_2^i_2⋯ t_d^t_d.

Each moment m_I is a homogeneous polynomial of degree | I |
 in the d^2+d unknowns x_kl.
This polynomial is multisymmetric:
it is invariant under permuting the vertices x_1,x_2,…,x_d+1.



This can be found in several sources, e.g. <cit.> and <cit.>.
 


Observe that the normalized moment generating function (<ref>)
is different from the  standard exponential moment generating function,
 commonly used in statistics and probability:

    ∑_i_1,i_2,…,i_d ∈m_i_1 i_2 ⋯ i_d/i_1 !   i_2 ! ⋯  i_d!   t_1^i_1 t_2^i_2⋯ t_d^t_d.

This is the exponential version of the ordinary generating function

    ∑_i_1,i_2,…,i_d ∈  m_i_1 i_2 ⋯ i_d
     t_1^i_1 t_2^i_2⋯ t_d^t_d.

The reason why we prefer (<ref>) over these is that
 (<ref>) and (<ref>) are not rational functions.
This can be seen already for d=1 and n=2 as  in Example <ref>.
In that case, (<ref>)
is the rational function in (<ref>), whereas
the two other series (<ref>) and (<ref>) are 
the non-rational functions

    ∑_i=0^∞m_i/i ! t^i  =  exp(bt) -  exp(at)/(b-a) t     and    ∑_i=0^∞ m_i t^i  =   log(1-ta)-  log(1-tb)/(b-a)t .





Let P be a full-dimensional simplicial  polytope in ^d
with vertices x_1,…,x_n, where n ≥ d+2.
Fix any triangulation Σ of P that uses only these vertices.
We identify Σ with the collection of subsets
σ = {σ_0,…,σ_d} that index the maximal simplices
conv(x_σ_0, …,x_σ_d). The volume
of P equals the sum of the volumes of these simplices. We write this as

    vol(P)    =  ∑_σ∈Σ vol(σ).

If μ_σ denotes the uniform probability distribution on each simplex σ then we have

    μ_P   =  1/ vol(P)∑_σ∈Σ vol(σ) μ_σ.

Since the moments depend linearly on the measure, 
Lemma <ref> implies the following result:

 
The normalized moment generating function for the uniform probability distribution  μ_P  
on the simplicial polytope P is equal to

    1/ vol(P)∑_σ∈Σ vol(σ)/∏_k ∈σ (1- x_k 1 t_1 - x_k 2 t_2 - ⋯ - x_k d t_d)     =   ∑_i_1,…,i_d ∈(i_1 +⋯+ i_d +d)!/i_1 !  ⋯  i_d!  d !
    m_i_1  ⋯ i_d t_1^i_1⋯ t_d^t_d.

This expression is independent of the triangulation Σ. The coefficient m_I = m_I(X) of  t^I
is a rational function whose numerator is a homogeneous polynomial of degree | I | + d in X and whose
denominator equals vol(P), which is a homogeneous polynomial of degree d in X.


To highlight the complexity of these moments, we examine the smallest non-simplex case.

[d=2, n=4] 
The polytope P is a quadrilateral in the plane, with
cyclically labeled vertices x_1,x_2,x_3,x_4. The
moments of its uniform probability distribution μ_P are rational functions in eight unknowns 
x_kl.
The area of the quadrilateral is the quadratic form

    vol(P)   =  1/2( x_11 x_22- x_12 x_21 
    + x_21 x_32- x_22 x_31 + x_31 x_42- x_32 x_41
     +  x_41 x_12- x_42   x_11 ).

The mean vector of the distribution μ_P is the centroid 
(m_10, m_01) = 1/2  vol(P)(M_10, M_01), where

    M_10    =        
    (x_41-x_21) (x_41+x_11+x_21) x_12  +  
    (x_11-x_31) (x_11+x_21+x_31) x_22  + 
          
    (x_21-x_41) (x_21+x_31+x_41) x_32  +    
    (x_31-x_11) (x_31+x_41+x_11) x_42,  
    
     M_01    =      
    (x_22-x_42) (x_42+x_12+x_22)x_11  +   
    (x_32-x_12) (x_12+x_22+x_32)x_21  +  
          
    (x_42-x_22) (x_22+x_32+x_42)x_31  +   
    (x_12-x_32) (x_32+x_42+x_12)x_41.

The covariance matrix of the distribution μ_P equals

    [ m_20 m_11; m_11 m_02 ]   =   1/24 · vol(P)·[ 2 M_20   M_11;   M_11 2 M_02 ],

where
 
    M_20    =    
    x_12(x_41-x_21)(x_11^2+x_11x_21+x_11x_41+x_21^2+x_21x_41+x_41^2)  + 
           
    x_22(x_11-x_31)(x_11^2+x_11x_21+x_11x_31+x_21^2+x_21x_31+x_31^2)  + 
           
    x_32(x_21-x_41)(x_21^2+x_21x_31+x_21x_41+x_31^2+x_31x_41+x_41^2)	 + 
           
    x_42(x_31-x_11)(x_11^2+x_11x_31+x_11x_41+x_31^2+x_31x_41+x_41^2).

The other diagonal entry M_02 is similar, and the off-diagonal entry equals

    M_11    =     
     (x_11x_22-x_12x_21) (2x_11x_12+x_11x_22+x_12x_21+2x_21x_22)   + 
           
     (x_21x_32-x_22x_31) (2x_21x_22+x_21x_32+x_22x_31+2x_31x_32)  + 
           
     (x_31x_42-x_32x_41) (2x_31x_32+x_31x_42+x_32x_41+2x_41x_42) + 
           
     (x_12x_41-x_11x_42) (2x_11x_12+x_11x_42+x_12x_41+2x_41x_42).

In Section <ref> we shall examine the relations satisfied by higher moments of quadrilaterals.


Let us return to Theorem  <ref> and take a closer look
at the rational function seen there. The normalized moment generating function 
can be written with a common denominator

    Ad_P( t_1,t_2,…,t_d)/∏_k=1^n (1- x_k1 t_1 - x_k2 t_2 - ⋯ - x_kd t_d).

The numerator is an inhomogeneous polynomial of degree at most n-d-1 in the variables t_1,t_2,…, t_d. Its coefficients are rational
functions in the entries of the n × d matrix X = (x_kl):

    Ad_P( t_1,t_2,…,t_d)    =    ∑_σ∈Σ vol(σ)/ vol(P)∏_k ∉σ(1- x_k1 t_1 - x_k2 t_2 - ⋯ - x_kd t_d),

where Σ is any triangulation of the simplicial polytope P.
Since (<ref>) does not depend on the triangulation 
Σ, so does the polynomial Ad_P. It is
an invariant of the simplicial polytope P.

We refer to Ad_P as the adjoint of P. This polynomial
was introduced by Warren to study barycentric coordinates in
geometric modeling <cit.>. He associates this
to the simple polytope P^* dual to P. For simplicity, we  assume 0 ∈ int(P).
The polytope P^* is the set of points (t_1,…,t_d) for which 
all linear factors in (<ref>) and (<ref>) 
are nonnegative. This implies that Ad_P is nonnegative on P^*.
The main result in <cit.> states that the adjoint depends only on P,
 and not on its triangulation Σ.
For us, this is a corollary to Theorem <ref>.
 
  The adjoint Ad_P is independent of the triangulation Σ of the polytope P.


The n linear factors in (<ref>),
(<ref>) and (<ref>) vanish on
the n facets of the dual polytope P^*. 
This imposes interesting vanishing conditions on the adjoint Ad_P.
A non-face of P is any subset  τ of {1,2,…,n} such that
 { x_k  :  k ∈τ} is not the vertex set of a face of P.
For any non-face τ, we write
L_τ for the affine-linear space in ^d that is
defined by the equations ∑_j= 1^d x_kj t_j = 1 for k ∈τ.
The collection of subspaces L_τ is denoted by 
𝒩ℱ(P). We call this the non-face subspace arrangement of
the simplicial polytope P. Equivalently, 𝒩ℱ(P) is the set 
of all intersections in ^d \ P^*
of collections of facet hyperplanes of the simple polytope P^*.


The adjoint Ad_P is a polynomial of degree at most n-d-1 that vanishes on the non-face subspace arrangement 𝒩ℱ(P).

 

The vanishing property follows from the fact that, for every non-face
τ of the polytope P, there exists a triangulation Σ of P
that does not have τ as a face.


In an earlier version of this article, 
we conjectured that, for every simplicial d-polytope P with n vertices,
the adjoint Ad_P is the unique polynomial 
of degree n-d-1 with constant term 1 that  vanishes on the non-face subspace arrangement 𝒩ℱ(P).
This is not quite true: 
For instance, if P is a regular octahedron 
such that
its three diagonals intersect in a common point (δ_1, δ_2, δ_3) ∈^3,
then the adjoint A_P is (1-δ_1t_1 - δ_2t_2 - δ_3t_3)^2 
and the non-face subspace arrangement 𝒩ℱ(P) consists of three lines in the plane defined by δ_1t_1 + δ_2t_2 + δ_3t_3 = 1.
So there is not a unique quadratic polynomial vanishing along 𝒩ℱ(P),
as every reducible quadratic polynomial with (1-δ_1t_1 - δ_2t_2 - δ_3t_3) as one of its two factors satisfies this vanishing property.
However, varying the vertices of P without changing its combinatorial type makes the three lines in 𝒩ℱ(P) skew
such that there is indeed a unique quadric surface passing through these three lines.
A corrected version of our conjecture was recently proven:

 
Let P be a d-polytope with n vertices.
If the projective closure ℋ_P^∗⊂^d
of the hyperplane arrangement formed by the linear spans of the facets of the dual polytope P^∗ is simple 
(i.e. through any point in ^d pass at most d hyperplanes in ℋ_P^∗),
there is a unique hypersurface in ^d of degree n-d-1 which vanishes along the projective closure of 𝒩ℱ(P).
The defining polynomial of this hypersurface is the adjoint of P.


We note that the assumption in Theorem <ref> that the hyperplane arrangement ℋ_P^∗ is simple implies that the polytope P is simplicial. 
For instance, for a regular octahedron P, the plane arrangement ℋ_P^∗ is not simple, but varying the vertices of P makes ℋ_P^∗ simple.

The adjoint Ad_P is closely related to 
barycentric coordinates on the simple polytope P^* and the associated
Wachspress variety in ^n-1; see <cit.>.
These objects can be defined as follows.
Suppose that the origin 0 lies in the interior of our simplicial polytope P,
and let Σ_0 be the triangulation of P
obtained by connecting 0 to the boundary of P.
The facets of Σ_0 are σ = 0 ∪ρ where 
ρ is any facet of P. The formula (<ref>)
holds for Σ_0, and we get

    Ad_P( t_1,t_2,…,t_d)    =    ∑_ρ  is  a  facet  of  Pβ_ρ∏_k ∉ρ(1- x_k1 t_1 - ⋯ - x_kd t_d).

Here β_ρ is the probability of the simplex 0 ∪ρ, which is given by 
| det( x_k : k ∈ρ)| divided by d  !   vol(P). 
Each summand in (<ref>) has degree n-d, but their 
sum has  degree n-d-1.

Let  N denote the number of facets ρ of P, i.e. the number of vertices of P^*.
Consider the map ^d →^N
whose coordinates are the following rational functions, one for each ρ:

    (t_1,…,t_d)  ↦  β_ρ∏_k ∉ρ(1- x_k1 t_1  - ⋯ - x_kd t_d)/ Ad_P( t_1,t_2,…,t_d) .

These are the barycentric coordinates of <cit.>.
 These coordinate functions 
are nonnegative on P^* and they sum up to 1. The image of P^* lies in the 
probability simplex with N vertices.   We call this the Wachspress model of P.
The term  model is meant in the sense of algebraic statistics <cit.>.
Its Zariski closure in  ^N-1 is the d-dimensional Wachspress variety of P.

In summary, the adjoint Ad_P  was introduced in geometric modeling by
Warren <cit.>. It equals the numerator of the normalized 
moment generating function for the uniform
distribution μ_P on a simplicial polytope P of type .
The map P ↦ Ad_P  represents the computation of all moments
of μ_P. This induces a polynomial map X ↦ Ad_X
on a dense open set of matrices X ∈^n × d.
 Its image lies in an
affine space of dimension n-1d - 1,
namely the space of polynomials of degree
n-d-1 in  d variables with constant term 1.
Passing to complex projective space, we define 
the adjoint moment variety
ℳ_ Ad() to be the Zariski closure
of this image in ^n-1d-1.
Readers of <cit.> are invited to regard ℳ_ Ad() as a moduli space
of Wachspress varieties, and to contemplate the questions in Section <ref>.





§ ONE-DIMENSIONAL MOMENTS



In this section we characterize the relations among the
moments of the 1-dimensional probability distributions that are obtained
by projecting the measures μ_P onto a line.
As before, let P be a d-dimensional simplicial polytope with n
vertices. We fix the coordinate projection π:  ^d →
that takes (t_1,t_2,…,t_d) to its first coordinate t_1.
The pushforward π_*(μ_P) is a probability distribution on the
line ^1. 
The ith moment m_i of this distribution equals the moment
m_i0⋯ 0 of μ_P. For normalized moment generating functions, equation
(<ref>) implies

    ∑_i=0^∞d+id m_i t^i    =   A_n-d-1(t)/ (1- u_1 t)(1- u_2 t) ⋯ (1-u_n t)  ,

where u_j = x_j1 is the first coordinate of the jth vertex of the polytope P,
and the numerator is A_n-d-1(t) =   Ad_P (t,0,0,…,0).
This is  a univariate polynomial of degree n-d-1.
We now confirm that the density of π_*(μ_P)
is the polytopal spline mentioned in the Introduction.


The density of π_*(μ_P) is a piecewise polynomial function
of degree d-1. Its value at any point a ∈^1 is the
(d-1)-dimensional volume of the fiber π^-1(a) ∩ P.
Moreover, this density function is d-2 times differentiable at its break points u_1, …, u_n.



The pushforward π_*(μ_P) is the measure that assigns
to a segment [v,w] in ^1 the nonnegative real number μ_P( π^-1([v,w]) ∩ P).
This number is the probability that a uniformly chosen random point in 
the d-polytope P has
its first coordinate between v and w. That probability can be computed
by integrating the normalized (d-1)-dimensional volumes of π^-1(a) for 
the scalars a  ranging from v to w.
It is well-known in the theory of polyhedral splines
(cf. <cit.>) that
this volume (called the polytopal density) is a piecewise polynomial function of degree d-1 in the 
parameter a.  This spline function is polynomial  on each of the intervals 
[u_i,u_i+1], and it is  d-2 times differentiable
at all its break points u_i.


Fix any integer r ≥ 2n-d and consider the moments 
m_0,m_1,…,m_r.
These correspond to the moments of μ_P whose index
set 𝒜 equals {{r }} = {i e_1 : i=1,2,…,r}.
Using the notation from the Introduction, we are interested in
the moment varieties ℳ_{{r}}() ⊂^r.


The moment variety ℳ_{{r }}() has dimension 
2n-d-1 in ^r. This variety depends only on d, n and r.
It is independent  of  the combinatorial type  of the polytope.



Consider the probability distribution π_*(μ_P) where
P runs over all polytopes of combinatorial type .
Such a distribution is parametrized by
the n parameters u_i in the denominator of 
(<ref>) and the n-d-1 nonconstant coefficients of the 
numerator polynomial A_n-d-1.
Thus there are 2n-d-1 degrees of freedom in specifying such a distribution,
or the associated spline function on ^1. Since the distribution
can be recovered from its first 2n-d moments (e.g. by <cit.>),
the irreducible variety ℳ_{{r}}() has dimension min(2n-d-1,r).

In the parametrization  above we obtain all polynomials
A_n-d-1 which are defined in some open set of the coefficient space ^n-d-1.
Hence the polytope type   imposes only inequalities
but no equations on that parameter space. We therefore conclude that, 
for any combinatorial type  of simplicial d-polytopes with n vertices,
the moment variety ℳ_{{r}}() is equal to the irreducible variety in ^r that is given by the 
parametric representation (<ref>).


We are now ready to state the main result in this section.
Our object of study is the subvariety 
ℳ_{{r}}(d,n) of ^r that is parametrically given by
 (<ref>), where u_1,u_2,…,u_n are
 arbitrary and A_n-d-1(t) ranges over polynomials 
with constant coefficient 1.
 We refer to this (2n-d-1)-dimensional variety as the
r-th moment variety of polytopal measures of type (d,n).
To describe its homogeneous prime ideal, we introduce the normalized moments 

    c_0 = c_1 = ⋯ = c_d-1 = 0      and     
    c_i+d = d+id m_i    for   i = 0,1, …, r.

We form
the following Hankel matrix
with n+1 rows and r+d-n+1 columns:

    (
    [       c_0       c_1         …       c_n     c_n+1         …   c_r+d-n;       c_1       c_2         …     c_n+1     c_n+2         … c_r+d-n+1;         ⋮         ⋮         ⋮         ⋮         ⋮;       c_n     c_n+1         …      c_2n    c_2n+1         …     c_r+d ]).

Note that  each entry of this matrix is a scalar multiple of one of the moments m_i.

  
  The homogeneous prime ideal in  ℝ[m_0, m_1, …, m_r] 
  that defines the moment variety ℳ_{{r}}(d,n) 
    is generated by the maximal minors of the Hankel matrix (<ref>).
    These minors form a reduced Gröbner basis with respect to any antidiagonal 
    term order, with initial monomial ideal ⟨ m_n-d,m_n-d+1,…,m_r-n⟩^n+1.
The degree of ℳ_{{r}}(d,n) equals r-n+d+1n.


The set-theoretic version of this theorem is implicit in the literature on polytopal moments
(cf. <cit.>). We offer a proof based on results from commutative algebra.


Let I be the ideal generated by the maximal minors of the matrix in (<ref>).
The statement that I is prime and has the expected codimension appears in <cit.>.
We fix the reverse lexicographic term order with
m_0 > m_1 > ⋯ > m_r. The leading monomial of each maximal minor
of (<ref>) is the product of the entries along the antidiagonal.
The ideal generated by all such antidiagonal products is the
(n+1)st power of the linear ideal ⟨ m_n-d,m_n-d+1,…,m_r-n⟩.
The codimension of that ideal equals the number r-2n+d+1
of occurring unknowns, and its degree is the number r-n+d+1n
of monomials of degree ≤ n in these unknowns.
The Gröbner basis property for that term order follows from
<cit.>. For an interesting refinement of that
Gröbner basis result see <cit.>.

It remains to show that our moment variety ℳ_{{r}}(d,n) equals the zero set of I.
Let M(t) denote the formal power series on the left-hand side of (<ref>).
Fix a polynomial β(t) = b_0 + b_1 t + b_2 t^2 + ⋯ + b_n t^n with unknown
coefficients such that β(t) M(t) is a polynomial of degree n-d-1. Hence the
coefficient of t^i in β(t) M (t) is zero for all integers i ≥ n-d.
This constraint is a linear equation in b = (b_n,b_n-1,…,b_1,b_0) whose
coefficients are the normalized moments c_j+d = d+jj m_j. More precisely,
the equation for the coefficient of t^i is

    b_n c_i+d-n + b_n-1 c_i+d-n+1 + ⋯ + b_2 c_i+d-2 + b_1 c_i+d-1 + b_0 c_i+d    =    0 .

These equations for i=n-d,n-d+1,…,r are equivalent to the requirement that
the row vector b is in the left kernel of the Hankel matrix (<ref>).
Hence that matrix has rank ≤ n on  ℳ_{{r}}(d,n). We conclude that 
  ℳ_{{r}}(d,n) is contained in the variety of I.
We already saw that both are irreducible varieties of the same dimension.
Therefore, they are equal.


 The recovery algorithm of <cit.> can be derived from the proof above.
For a given valid sequence of real moments, the Hankel matrix (<ref>)
has rank n. For such a matrix, we compute a generator
b = (b_n, …,b_1,b_0) of its left kernel.
The node points u_1,…,u_n are recovered as the roots of
β(t) = ∑_i=0^n b_i t^i. The numerator polynomial
in (<ref>) is found to be

    A_n-d-1(t)    =   1/b_0∑_ℓ = 0^n-d-1( ∑_i=0^ℓ b_i c_ℓ+d-i) · t^ℓ.



It is  instructive to revisit Example <ref>  from the perspective of Theorem <ref>.

[d=1,n=2] 
The variety ℳ_{{r}}(1,2) is the moment surface
in ^r whose points represent the uniform probability distributions
on line segments in ^1. The prime ideal of this surface
is generated by the 3 × 3 minors of the 3 × r Hankel matrix

    ([           0         m_0        2m_1        3m_2        4m_3           ⋯ (r-1) m_r-2;         m_0       2 m_1       3 m_2       4 m_3       5 m_4           ⋯     r m_r-1;       2 m_1       3 m_2       4 m_3       5 m_4       6 m_5           ⋯   (r+1) m_r;             ]).

These cubics form a Gröbner basis. The moment surface has degree r2 in ^r.
Up to a factor of 4, the leftmost 3 × 3 minor is equal to the cubic
 (<ref>)  whose surface is shown in Figure <ref>.

   
 

§ SIMPLICES

 
 
 In what follows we focus on the case n=d+1 when the
polytope P  is the d-simplex Δ_d with vertices
x_k = (x_k1,x_k2, …,x_kd) for k=1,…,d+1.
From the normalized moment generating function in Lemma <ref>
we can derive the following explicit formula
for the moments of μ_Δ_d.



For I = (i_1,…,i_d) ∈^d, the corresponding moment of the simplex equals

    m_I(X) 
       =   i_1 !  i_2 ! ⋯  i_d !   d!/(i_1+i_2+⋯ + i_d+d)!·∑_u∏_k=1^d+1(u_k1+ u_k2+⋯+ u_kd)!/u_k1! u_k2! ⋯ u_kd!
    x_k1^u_k1 x_k2^u_k2⋯ x_kd^u_kd,

where the sum is over nonnegative integer
(d+1)× d   matrices u with column sums given by I.


Proposition <ref> shows that m_I is a
fairly complicated polynomial of degree |I|
in the d^2+d entries of X=(x_1,…, x_d+1)^T.
However, these polynomials are still simpler than the rational functions
we obtain for moments of polytopes other than simplices.
For instance, consider the subalgebra of [X] generated
by all moments m_I(X) in (<ref>) where I runs over ^d.
We shall argue in Section <ref> that this is the algebra
of multisymmetric polynomials <cit.>.

 In this section we are interested in the polynomial relations
among the moments m_I where I runs over an appropriate
finite subset 𝒜 of ^d \{0}. We homogenize
these relations with the special unknown m_00 ⋯ 0 
that represents the total  mass of the simplex.
This gives us homogeneous polynomial relations among the moments
indexed by 𝒜∪{0}. Their zero set in ^|A|
is the moment variety ℳ_𝒜(Δ_d).
The special case d=1 and 𝒜 = {{r}},
where our variety is a surface,
was seen in Example <ref>.

We next present an algorithm for recovering the 
(d+1) × d matrix X =(x_kl) from the above moments m_I of order 
 |I| ≤ d+1. There are 2d+1d such moments m_I.
Let 𝕃 denote the sum of all terms 
on the right-hand side in (<ref>) where
1 ≤ i_1 + i_2 + ⋯ + i_d ≤ d+1.
This is a polynomial in t_1,t_2,…,t_d with zero constant term.
We compute the formal inverse:

    (1+ 𝕃)^-1   =    1 - 𝕃 + 𝕃^2 - 𝕃^3 +
    𝕃^4 +   ⋯  + (-1)^d+1𝕃^d+1   mod  ⟨ t_1,t_2,…,t_d ⟩^d+2.

Thus (1+𝕃)^-1 is a polynomial of degree ≤ d+1 in t_1,t_2,…,t_d
with constant term 1. This polynomial must  factor into linear factors,
one for each vertex of the desired simplex:

    (1+𝕃)^-1   =   ∏_k=1^d+1( 1  -  x_k1 t_1 - x_k2 t_2 - ⋯ - x_kd t_d ).

A necessary and sufficient condition for such a factorization 
to exist is that the coefficients of (1+𝕃)^-1 satisfy Brill's equations 
<cit.>.
These classical equations  characterize  polynomials
that are products of linear factors, among all
polynomials of degree ≤ d+1 in d variables.                   
We write [d+1] for the set of vectors I ∈^d with |I| ≤ d+1.
Our discussion implies:

 
Homogeneous equations that define
ℳ_[d+1](Δ_d) set-theoretically
are obtained by substituting the polynomials in m_I
on the left-hand side of (<ref>) into Brill's equations.

 
If we are given numerical values in  for the moments m_I
then the factorization (<ref>) is found in exact
arithmetic by the built-in factorization methods in
any computer algebra system, provided the vertex
coordinates x_kl of our simplex are rational numbers.
If the moments m_I are rational but the x_kl are not rational
then they are algebraic over ℚ, and one can use 
algorithms for absolute factorization to obtain the
right-hand side of (<ref>). If the moments are floating point
numbers then one uses tools from
numerical algebraic geometry
(e.g. the software Bertini <cit.>) to obtain
an accurate  factorization purely numerically.

We now return to the problem of computing the prime ideal
of our variety ℳ_[d+1](Δ_d).
In practise, the method in Corollary <ref> did not
work so well. In what follows, we discuss some
 techniques that we found more effective in obtaining relations among moments.

In all  computations, it helps to use the fact that
the ideal of ℳ_𝒜()
is homogeneous with respect to a
natural ^d+1-grading. 
On the unknown moments this grading is given by

    degree(m_i_1 i_2 ⋯ i_d)    =   
    (1,i_1,i_2,…,i_d).

This follows from the parametric representation 
of the moment varieties given in (<ref>).
Our first result concerns the case d=2, i.e., the ideal of a moment variety for triangles.


 
The triangle moment variety ℳ_[3](Δ_2)
has dimension 6 and degree 30. It lives in the projective space ^9.
Its prime ideal  is minimally generated by 
eight quartics and one sextic. 
The degrees of the nine ideal generators in the 
^3-grading given in (<ref>) are 

    (4,2,3), (4,3,2), (4,2,4), (4,3,3), (4,3,3), (4,4,2),  (4,3,4), (4,4,3), (6,6,6).




This computation was carried out with the technique of cumulants, to be introduced below.
For an explicit example, the  ideal generator of degree (4,2,3)   equals

    3 m_02 m_10^2 m_01 - 6 m_11 m_10 m_01^2 + 
    3 m_20 m_01^3-m_03 m_10^2 m_00
    +4 m_11^2 m_01 m_00 +m_21 m_02 m_00^2 
    
    - 4 m_20 m_02 m_01 m_00 
    + 2 m_12 m_10 m_01 m_00 -m_21 m_01^2 m_00
     +m_03 m_20 m_00^2-2  m_12 m_11 m_00^2 .

We shall present the derivation by means of Macaulay2 in the proof of Proposition <ref>.



Logarithms turn products into sums, and this can greatly
simplify calculations. To do this in the context of probability and
statistics, one passes from moments to cumulants.
Let 𝕄 be the generating function on the right of (<ref>).
The associated normalized cumulant generating function is defined
as the formal logarithm via log(1+x) = 
x- 1/2x^2 + 1/3 x^3 - ⋯:

    𝕂    =     log(𝕄)    =   ∑_i_1, …,i_d ∈(i_1 + i_2 +⋯+ i_d -1)!/i_1 !    i_2 !  ⋯   i_d!  k_i_1 i_2 ⋯ i_d t_1^i_1 t_2^i_2⋯ t_d^i_d.

Here k_00 ⋯ 0 = 0. By comparing the coefficients of monomials t^I in this identity,
we obtain the expressions for each cumulant k_I as a polynomial in the moments 
m_J where |J| ≤ |I|.  

[d=2] 
Here are the formulas for the cumulants k_I of order |I| ≤ 3 in terms
of the moments m_J of order |J| ≤ |I|, written
in the language of Macaulay2 <cit.>:



We shall revisit this piece of code shortly, to represent the ideal generators in Proposition <ref>.


The transformation (<ref>) from moments
to cumulants is easily invertible. Namely, the moment generating function is the
exponential of the cumulant generating function:

    𝕄     =      exp(𝕂)   =   
     1  + 𝕂 + 1/2𝕂^2  + 1/6𝕂^3  + 1/24𝕂^4   +   ⋯.

 This identity expresses each moment  m_I as a polynomial in the cumulants k_J 
 where |J| ≤ |I|.
     
     The factorial multipliers in the generating function (<ref>) 
     are chosen  so that the normalized cumulants 
     k_i_1 i_2 ⋯ i_d of a simplex Δ_d coincide with the standard 
    power sum multisymmetric functions <cit.>
     in its vertices x_1,…,x_d+1. 
     This  is the content of the following corollary.

          


The cumulants of the uniform probability distribution on the simplex Δ_d  are 


    k_i_1 i_2 ⋯ i_d   =   ∑_j=1^d+1
    x_j 1^i_1 x_j 2^i_2⋯ x_j d^i_d.

    


Taking the logarithm of the left-hand side in (<ref>), we
see that 𝕂 is the sum of the expressions
-  log(1 - x_j 1 t_1 - x_j 2 t_2 - ⋯ - x_j d t_d )
for j=1,2,…,d+1. The coefficient of a non-constant
monomial t_1^i_1 t_2^i_2⋯ t_d^i_d in 
the expansion of that expression equals
x_j 1^i_1 x_j 2^i_2⋯ x_j d^i_d.


 Both the moments (<ref>) and the
cumulants (<ref>) are multisymmetric functions in x_1,x_2,…,x_d+1,
and they are expressible in terms of each other.  However, the
formula for the cumulants is much simpler than that for the moments.
For that reason, it seems advantageous to use cumulant coordinates
when studying the moment varieties of simplices.


 Replacing moments with cumulants amounts to a change
 of coordinates in the affine space
 
    𝔸^d+rd-1   =   { m_00 ⋯ 0 = 1 }   =   { k_00 ⋯ 0 = 0 }.

This is the affine chart of interest inside the projective space 
 (<ref>) which harbors ℳ_[r](Δ_d).
 
 Ciliberto et al. <cit.>  refer to this non-linear automorphism
    as a Cremona linearization.
In our situation, the Cremona linearization greatly simplifies the equations
that define ℳ_[r](Δ_d). We now illustrate this explicitly
for a simple case, namely for triangles (d=2) with k=3.
Here, Cremona linearization identifies our moment variety with a Grassmannian.

 
The restriction of the  6-dimensional triangle moment variety ℳ_[3](Δ_2)
in  ^9 to  the affine chart 𝔸^9 = {m_00 = 1 } = {k_00 = 0} 
can be identified with an affine chart of the 6-dimensional Grassmannian of lines in ^4,
which  has its  Plücker embedding in ^9.



 We give the identification with the Grassmannian as
 Macaulay2 code, starting from Example <ref>.
 The following ten expressions in the cumulants serve
as Plücker coordinates:

We next form the ideal generated by the five quadratic Plücker relations:

The ideal I now contains five of the eight quartics in Proposition <ref>
starting with that of degree (4,2,3) in (<ref>).
These five quartics generate the prime ideal of
the affine variety ℳ_[3](Δ_2) ∩𝔸^9.
To pass to the projective closure in ^9 we now homogenize and saturate:

This displays all nine minimal generators of the homogeneous prime ideal  of ℳ_[3](Δ_2).


Using the cumulant coordinates, it is possible to derive defining equations for
ℳ_[r](Δ_d) for r ≥ d+2 in terms of the equations for r=d+1.
This is done by the following technique:

 
For the uniform probability distribution on the simplex Δ_d,
each cumulant k_I of order |I| ≥ d+2  is a polynomial 
in the cumulants k_J of order |J| ≤ d+1.



We abbreviate X_k = x_k1 t_1 + x_k2 t_2 +  ⋯ + x_kd t_d for k=1,2,…,d+1.
For any ℓ≥ d+2, we consider the power sum X_1^ℓ + X_2^ℓ + ⋯ + X_d+1^ℓ.
Using Newton's identities, we can write this uniquely as a polynomial P_ℓ with rational coefficients in
the first d+1 such power sums:

    ∑_k=1^d+1 X_k^ℓ     =      P_ℓ( ∑_k=1^d+1 X_k^1 , ∑_k=1^d+1 X_k^2  , …, ∑_k=1^d+1 X_k^d+1).

By Corollary <ref>, the left-hand side is the following polynomial of degree ℓ in t_1,…,t_d:

    ∑_k=1^d+1 X_k^ℓ     =     ∑_I  :  |I| = ℓ|I|I  k_I  t^I .

The same holds for the power sums occurring on the right-hand side of (<ref>).
We expand the right-hand side and write it as a polynomial in t_1,t_2,…,t_d.
Then each coefficient is a polynomial in the cumulants k_J with |J| ≤ d+1.
This gives the desired formula for k_I.


[d=2] The five fourth-order cumulants for a triangle in the plane ^2 admit the following 
polynomial expressions in terms of  the nine cumulants of lower order:


    k_04   =     k_01^4+3 k_02^2+4 k_03 k_01 - 6 k_02 k_01^2, 
    
    k_13   =    -3 k_02 k_10 k_01+3 k_11 k_02
    +k_03 k_10+ 3 k_12 k_01-3 k_11 k_01^2+k_10 k_01^3 ,
    
    k_22   =    k_20 k_02  -k_02 k_10^2+2 k_11^2+2 k_12 k_10
    +k_10^2 k_01^2+2 k_21  k_01-4 k_11 k_10 k_01 -k_20 k_01^2 , 
    
    k_31   =    k_30 k_01+3 k_20 k_11+3 k_21 k_10
    - 3 k_11 k_10^2 - 3 k_20 k_10 k_01 +k_10^3 k_01, 
    
    k_40   =    k_10^4+3 k_20^2+4 k_30 k_10-6k_20 k_10^2.

These identities hold if we substitute
 k_ij  =   x_11^i x_12^j  +   x_21^i x_22^j  +  x_31^i x_32^j, so 
 they provide valid equations
 for ℳ_[4](Δ_2) on the affine chart 𝔸^14 = {m_00  = 1}.
 To translate these equations into moment coordinates, we simply use the identities
 arising from 𝕂  =   log(𝕄), such as
 

    k_04   =    60m_04- 72 m_02^2-81 m_01^4+216 m_01^2 m_02 -120 m_01 m_03, 
    
    k_13    =     60 m_13+ 108 m_10 m_01 m_02- 30 m_10 m_03- 81 m_10 m_01^3 + 108 m_01^2 m_11- 90 m_01 m_12- 72 m_11 m_02.

Consider the ideal generated by these polynomials in moments.
Just like in the end of the proof of Proposition <ref>, we 
homogenize and saturate  with respect to m_00. This yields
generators for the homogeneous prime ideal of
the triangle moment variety ℳ_[4](Δ_2) in ^14.


At this point, we note that all results in this section are valid
for configurations of n ≥ d+2 points x_1,x_2,…,x_n in ^d,
but with the uniform measure on their convex hull replaced by a canonical polytopal measure.
Namely, consider the generating function on the left-hand side in (<ref>)
but with the upper index n instead of d+1.
This is the normalized moment generating function for the probability measure
π_*(μ_Δ_n-1) on ^d where π denotes the canonical projection
from the simplex Δ_n-1 onto the polytope P  =  conv(x_1,x_2,…,x_n):

    ∏_k=1^n1/1 - (x_k1 t_1 + x_k2 t_2 +⋯+ x_kd t_d)    =  ∑_i_1,i_2,…,i_d ∈(i_1 + i_2 +⋯+ i_d +n-1)!/i_1 !   i_2 ! ⋯  i_d!  (n-1) !
    m_i_1 i_2 ⋯ i_d t_1^i_1 t_2^i_2⋯ t_d^t_d.

The density function of
π_*(μ_Δ_n-1) is the canonical polytopal spline  supported on P.
This is piecewise polynomial of degree n-d-1 and differentiable of order n-d-2  <cit.>.

We consider the moments m_I of order |I | ≤ r on the right-hand side above. These are polynomial functions
in the nd unknowns x_kl.
Let  ℐ_r,d,n denote the prime ideal of homogeneous polynomial relations
among these r+dd moments. For instance, 
the ideal ℐ_3,2,3 is the one with 9 
generators in 10 unknowns seen  in Propositions <ref>
and <ref>.

It would be interesting to compute the ideals ℐ_r,d,n
for as many values of r, d and n as possible, and to better understand their varieties.
For instance, the case d=2 and n=4
concerns the canonical piecewise linear densities on quadrilaterals in ^2.
It should be compared to the uniform distribution on  quadrilaterals,
to be studied in Section <ref>.



We conclude this section with a discussion of the tetrahedron Δ_3. This has 12 parameters,
namely the coordinates of the vertices x_k = (x_k1,x_k2,x_k3) for k=1,2,3,4.
We are interested in the moment variety ℳ_[3](Δ_3) in 
^19. Points on this variety represent  cubic surfaces in ^3.
The coefficients of a cubic are specified by
the cumulants of the uniform distribution on Δ_3:

    k_ijl   =   
    x_11^i x_12^j x_13^l  + 
    x_21^i x_22^j x_23^l  + 
    x_31^i x_32^j x_33^l  + 
    x_41^i x_42^j x_43^l 
        for 1 ≤ i+j+l ≤ 3.

We computed polynomials in the prime ideal of relations among these 19 cumulants. 
This ideal is not homogeneous in the usual grading but it is homogeneous
in the ^3-grading given by deg(k_ijl) = (i,j,l). For a concrete example,
here is an ideal generator of degree (3,2,2):
  
    k_010^2 k_200 k_102+k_100^2 k_020 k_102+k_001^2 k_100 k_110^2+k_010^2 k_100 k_101^2
    -k_100^3 k_002 k_020+k_100^2 k_002 k_120+k_100^3 k_011^2+4 k_110^2 k_102
    +2 k_001 k_100^2 k_020 k_101
    
    +4  k_101^2 k_120+4 k_011^2 k_300
    -2 k_001 k_010 k_100 k_101 k_110+2 k_001 k_010 k_100 k_011 k_200
    -k_010^2 k_100 k_002 k_200-k_001^2 k_100 k_020 k_200-2 k_010 k_100^2 k_011 k_101
     +2 k_010 k_100^2 k_002 k_110
     -2 k_001 k_100^2 k_011 k_110 
    +10 k_100 k_011 k_101 k_110+5 k_100 k_002 k_020 k_200-2 k_010 k_100 k_110 
    k_102 
    +2 k_010 k_100 k_101 k_111
     +2 k_001 k_100 k_110 k_111 
    -2 k_001 k_010 k_200 k_111 
    -2 k_001 k_100 k_101 k_120+2 k_010 k_100 k_011 k_201-2 k_001 k_100 k_020 k_201
    +2 k_001 k_010 k_110 k_201
     -2 k_010 k_100 k_002 k_210
     +2 k_001 k_100 k_011 k_210
    +2 k_001 k_010 k_101 k_210-2 k_001 k_010 k_011 k_300+k_001^2 k_200 k_120
    -4 k_002 k_020 k_300+8 k_002 k_110 k_210
     -8 k_101 k_110 k_111 
     -2 k_100^2 k_011 k_111
    -4 k_002 k_200 k_120+8 k_011 k_200 k_111-5 k_100 k_002 k_110^2-5 k_100 k_020 k_101^2 
    +k_010^2 k_002 k_300 -4 k_020 k_200 k_102
     -5 k_100 k_011^2 k_200+8 k_020 k_101 k_201
    -8 k_011 k_110 k_201-8 k_011 k_101 k_210+k_001^2 k_020 k_300
    -2 k_001^2 k_110 k_210-2 k_010^2 k_101 k_201.
  
Each relation among cumulants translates into a ^4-homogeneous relation among
the moments. The above polynomial translates into the
following polynomial of degree (5,3,2,2):


    m_000^2 m_002 m_020 m_300-2 m_000^2 m_002 m_110 m_210
    +m_000^2 m_002 m_120 m_200
    -m_000^2 m_011^2 m_300+2 m_000^2 m_011 m_101 m_210
    +2 m_000^2 m_011 m_110 m_201
    
    -2 m_000^2 m_011 m_111 m_200-2 m_000^2 m_020 m_101 m_201+m_000^2 m_020 
    m_102 m_200
    -m_000^2 m_101^2 m_120+2 m_000^2 m_101 m_110 m_111 
    -m_000^2 m_102 m_110^2 
    
     -2 m_000 m_001 m_010 m_110 m_201
     +2 m_000 m_001^2 m_110 m_210 
    -m_000 m_001^2 m_120 m_200
    +2 m_000 m_001 m_010 m_011 m_300 -2 m_000 m_001 m_010 m_101 m_210
     -m_000 m_001^2 m_020 m_300
    +2 m_000 m_001 m_010 m_111 m_200
    -2 m_000 m_001 m_011 m_100 m_210+2 m_000 m_001 m_020 m_100 m_201 
    +2 m_000 m_001 m_100 m_101 m_120
    
    -2 m_000 m_001 m_100 m_110 m_111
    -m_000 m_002 m_010^2 m_300+2 m_000 m_002 m_010 m_100 m_210-5 m_000 m_002 m_020 m_100 m_200 
    -m_000 m_002 m_100^2 m_120
     +2 m_000 m_010 m_100 m_102 m_110
    +2 m_000 m_010^2 m_101 m_201 
    -m_000 m_010^2 m_102 m_200-2 m_000 m_010 m_011 m_100 m_201
     -2 m_000 m_010 m_100 m_101 m_111
    
    +5 m_000 m_002 m_100 m_110^2
    +5 m_000 m_011^2 m_100 m_200+2 m_000 m_011 m_100^2 m_111
    -10 m_000 m_011 m_100 m_101 m_110-m_000 m_020 m_100^2 m_102
    
    +5 m_000 m_020 m_100 m_101^2 
    +4 m_001^2 m_020 m_100 m_200-4 m_001^2 m_100 m_110^2-8 m_001 m_010 m_011 m_100 m_200
    +8 m_001 m_010 m_100 m_101 m_110
     +8 m_001 m_011 m_100^2 m_110
     -8 m_001 m_020 m_100^2 m_101 
    +4 m_002 m_010^2 m_100 m_200-8 m_002 m_010 m_100^2 m_110+4 m_002 m_020 m_100^3 
    
    -4 m_010^2 m_100 m_101^2+8 m_010 m_011 m_100^2 m_101-4 m_011^2 m_100^3.



Based on our computations, we propose the following conjecture.


Consider cumulants and moments of order ≤ 3 for the uniform distribution
on a tetrahedron. They specify irreducible varieties of
dimension 12 in 𝔸^19 and ^19 
respectively. The prime ideal for  cumulants has 44 minimal generators. Their degrees are


    (223),(232),(322),
    (134),(143),(314),(341),(413),(431),
    (224),(224),(242),(242),(422),(422),
    (332),(332),(332),(323),(323),(323),(233),
    (233),(233), 
    (144),(414),(441),(333),(333),(225),(252),(522),
    (234),(234),(243),(243),(324),(324),
    (342),(342),(423),(423),(432),(432).


The prime ideal for moments has 93 minimal generators, namely
90 quintics and 3 sextics.


We shall return to the 90 ideal generators of degree five in Proposition <ref>.



§ SYMMETRY AND INVARIANTS



In this section we study the symmetries arising from the  group 
of affine transformations:

    Aff_d   :=   ℝ^d ⋊GL_d().

This group is a subgroup of GL_d+1(). It acts on column vectors x = (x_1,…,x_d)^T via

    [ x; 1 ]  ↦  [ A b; 0 1;   ][ x; 1;   ],

where A = (a_ij) is an invertible d × d-matrix and
b = (b_i) is a column vector in ^d.
This group acts naturally on the space of
realizations of a polytope type 𝒫.
The action (<ref>)
also induces an action on monomials and hence an action on moments m_I, I ∈^d.
Explicitly, 

    m_I  ↦  ∑_Jν_IJ· m_J,

where ν_IJ = ν_IJ(A,b) is the coefficient of 
the monomial x^J in the expansion of (A x + b)^I.
The sum in (<ref>) is over all J ∈^d
such that |J| ≤ |I|. Here are formulas for two simple cases.

[d=1] The group Aff_1 acts on the real line ^1 via x ↦ ax + b,
where a,b ∈ with a ≠ 0. Under this action, the i-th moment 
of a probability measure on ^1 is transformed into the following linear combination
of all moments of order at most i:

    m_i   ↦  ∑_j=0^i ij a^j b^i-j m_j.



[d=2] The moments of order ≤ 2 are the entries of
the symmetric matrix

    M  =  [ m_20 m_11 m_10; m_11 m_02 m_01; m_10 m_01 m_00 ].

The upper left 2 × 2 block is the covariance matrix.
The group Aff_2 consists of 3 × 3 matrices

    Ab    =   [ a_11 a_12  b_1; a_21 a_22  b_2;    0    0    1 ].

With this matrix notation for |I| ≤ 2, the action (<ref>)   takes the form
M  ↦  Ab · M · Ab^T.
More generally, if we consider all moments of order ≤ r then
we can write these as an 
r-dimensional symmetric tensor of format 3 × 3 ×⋯× 3.
The action (<ref>) is given by multiplication of this tensor on all its r sides
by the 3 × 3 matrix Ab. 


We have identified the space of moments of order ≤ r with the projective space
^d+rd-1.
The formula (<ref>) defines a  linear action
of the group Aff_d on that projective space. 
Recall that, for each simplicial polytope
P in ^d and each subset 𝒜⊂ℕ^d with 0 ∉𝒜, 
its associated moment variety ℳ_𝒜()
is a projective variety in ^|𝒜|.
In particular, if 𝒜 is the index set
[r]  = { I ∈ℕ^d	| 1 ≤ |I| ≤ r }, 
then the moment variety ℳ_[r]() lives in ^d+rd-1,
as in (<ref>).



The moment variety ℳ_[r]() of
a simplicial polytope P in ^d is invariant under the action of the group Aff_d
of affine transformations on  the projective space ^d+rd-1.



The group Aff_d acts on ^d+rd-1, and it
also acts on the space of all realizations P of a given  combinatorial type .
The map that takes a specific simplicial polytope P to its point in the
variety ℳ_[r]() is equivariant with respect to
the two actions, i.e. the image of P under an affine transformation
is mapped to the image of its moment vector under the same transformation.
This implies that ℳ_[r]()  is invariant under 
the action by Aff_d.












In cases where our moment variety is a hypersurface 
in ^d+rd-1,  its defining equation 
is a polynomial  that is invariant under Aff_d.
It is thus of interest to study the invariant ring

    [  m_I  :  |I| ≤ r  ]^ Aff_d.

 Here and in what follows we use the term invariant  for
relative invariants, i.e. such that the transformation of an invariant polynomial equals the
original polynomial times a power of det(A). In other words,
an invariant is an absolute invariant of the subgroup Aff_d ∩ SL_d+1().

[d=1,r=3]  The group Aff_1 acts on
the polynomial ring [m_0,m_1,m_2,m_3] via (<ref>).
The invariant ring has four generators, in degrees
(1,1), (2,2), (3,3) and (4,6):

    a = m_0  ,   
    b = m_0 m_2-m_1^2   ,   
    c = m_0^2 m_3-3 m_0 m_1m_2+2 m_1^3, 
    
    d  =  m_0^2 m_3^2-6m_0 m_1 m_2 m_3+4 m_0 m_2^3+4 m_1^3 m_3-3 m_1^2 m_2^2.

We verified the equality [m_0,m_1,m_2,m_3]^ Aff_1 = [a,b,c,d], see  Theorem <ref>
below.
Note that b and d are the discriminants of binary forms of degree two and three.  The four invariants
satisfy the relation a^2 d - 4 b^3 - c^2  =   0. This expresses the discriminant
d in terms of the other three invariants on the affine open set {m_0 = 1}.
The invariant c is the one of interest to us. The cubic surface it
defines in ^3 is given by (<ref>) and shown in 
Figure <ref>.


Once we know generators for this invariant ring (<ref>),
 we can try to express our hypersurface
as a polynomial in these fundamental invariants.
Note that Hilbert's theorem on finite generation
does not directly apply  here, because
the group Aff_d is not reductive.
However, there is a nice method from
classical invariant theory using covariants, which allows us to conclude
finite generation and to compute the invariants of Aff_d we are interested in.

Set V = ^d+1, with standard basis denoted by {u_1,u_2,…,u_d+1}. 
We identify the symmetric power S_r(V) with our
space of moments m_I of order |I| at most r.
The action of the general linear group G =  GL_d+1() on
   S_r(V) restricts to the action of the affine group Aff_d on moments.
 The group G acts naturally on the dual space V^*. 
 We consider the action of G on the direct sum
S_r (V) ⊕ V^*, and the induced action on the polynomial ring

    [ S_r(V) ⊕ V^*]    =    [ m_I  :  |I| ≤ r ]  ⊗_ [ u_1,u_2,…,u_d+1 ].

A G-invariant in this polynomial ring is known as a covariant.
Thus [ S_r(V) ⊕ V^*]^G is the ring of covariants of
S_r(V). This ring is finitely generated because G is reductive.

Let ψ be the ring epimorphism [S_r(V) ⊕ V^*]  →[ S_r(V) ]
defined by u_d+1↦ 1 and u_i ↦ 0 for i =1,2,…,d.
This reflects the special role played by the last index 
in the realization of Aff_d as a subgroup of G.
The following result is known in classical invariant theory.

 
The map ψ induces an isomorphism between covariants and affine invariants:

    [ S_r(V) ⊕ V^* ]^G   ≃  [ S_r(V) ]^ Aff_d.




This statement is a special case of <cit.>. 


The  basic covariant is the homogeneous polynomial  itself. In our notation,

    f  =  f(m,u)      =    ∑_|I| ≤ rrI, r-|I|· m_I · u^I u_d+1^r-|I|.

The image of f under the isomorphism (<ref>) is the
 Aff_d-invariant moment coordinate 
    ψ(f) = m_00 ⋯ 0.

The degree of a covariant g = g(m,u) is its degree in the unknowns m_I.
The order of g is its degree in the unknowns u_j.
The form f is a covariant of degree 1 and order r.
Covariants of order 0 are invariants of G. The degree of an affine invariant 
in the ^d+1-grading (<ref>) can be
read off from the degree and the order of the corresponding covariant:


Let g be a covariant of degree p and order o for the space S_r(^d+1) 
of degree r forms. Then the integer rp-o is a positive multiple of 

the number d+1 of unknowns.  Setting q =rp-o/d+1,
the degree of the associated affine invariant ψ(g) equals
(p,q,q,…,q).
 

 
Consider the diagonal matrix diag(t,t,…,t)
in G =  GL_d+1(). It acts on S_r(V) 
by mutiplying the vector m with t^r. It acts on V^*
by multiplying the vector u with t^-1. The covariant g(m,u) of
degree p and order o is transformed  by the action of this diagonal matrix into
g(t^r m, t^-1 u) = t^pr-o g(m,u).
The multiplier t^pr-o is a power of
det(A) = t^d+1, so q = rp-o/d+1 is an integer.
It follows that diag(t_1,t_2,…,t_d+1)
takes g(m,u) to t_1^q t_2^q ⋯ t_d+1^q g(m,u).
We find that  ψ(g)(m) = g(m,e_d+1) has degree (p,q,…,q)
with respect to the grading in (<ref>).


[d=1,r=3] We derive Example <ref> from the
classically known covariants of the binary cubic. The four generators 
of the ring [m_0,m_1,m_2,m_3,u_1,u_2]^G are


  * the binary cubic  A itself, of degree 1 and order 3, 


  * the Hessian B, of degree 2 and order 2, 


  * the Jacobian of A and B, denoted by C, of degree 3 and order 3, 


  * the discriminant D, of degree 4 and order 0.

Applying ψ to these covariants yields the corresponding affine invariants in
Example <ref>.


[d=2,r=3]  Consider any  probability measure on ^2.
Its moments of order ≤ 3  can be encoded as the coefficients of a ternary cubic

    f      =    
    m_30u_1^3+3 m_21 u_1^2 u_2 + 3 m_20 u_1^2 u_3
    +3 m_12 u_1 u_2^2 + 6 m_11 u_1 u_2 u_3 
           +3 m_10 u_1 u_3^2
    +m_03 u_2^3 + 3 m_02 u_2^2 u_3
    +3 m_01 u_2 u_3^2 + m_00 u_3^3.

The notation is as in (<ref>).
It is  classically known that f has six fundamental covariants:

    covariant     f      S     T       H     G     J 
    
    ( degree,  order )    (1,3)     (4,0)     (6,0)     (3,3)     (8,6)     (12,9)

First is the ternary cubic f itself, of degree 1 and order 3.
Next are the  Aronhold invariants S and T, of degree 4 and 6 resp. 
These are followed
by the Hessian H.
The covariant G is explained in Dolgachev's book <cit.>,
where the following formula can be  found:

    G    =    det[ f_11 f_12 f_13  h_1; f_12 f_22 f_23  h_2; f_13 f_23 f_33  h_3;  h_1  h_2  h_3    0 ]    with  f_ij = ∂^2 f/∂ u_i ∂ u_j
    and h_i = ∂ H/∂ u_i.

The last covariant J is the Jacobian of f, H and G.
This is known as the Brioschi covariant.

The six fundamental affine invariants are the images of the fundamental covariants 
under replacing (u_1,u_2,u_3) with (0,0,1):

    affine invariant    m_00 =ψ(f)      s = S      t = T     h = ψ(H)      g = ψ(G)     j = ψ(J) 
    ^3-degree    (1,0,0)      (4,4,4)     (6,6,6)     (3,2,2)     (8,6,6)     (12,9,9) 
    # terms     1     25     103     5     168      892

 

We summarize our derivation of the affine invariants of ternary cubics as follows:

 
For d=2 and r=3, the invariant ring (<ref>)
equals [m_00,s,t,h,g,j] modulo one
homogeneous relation of degree
(24,18,18). Hence its Hilbert series equals

    1 + x^12 y^9 z^9/
    (1-x)(1-x^4 y^4 z^4) (1- x^6 y^6 z^6)(1-x^3 y^2 z^2) (1-x^8 y^6 z^6) (1-x^12 y^9 z^9) .



The moment varieties ℳ_[r]() are hypersurfaces in only very few cases.
Examples include
=  quadrilateral with r=3, 
= 13-gon with r=6,  or = octahedron with r=3.
In those cases there is a single affine invariant.
The first one is featured in the next section.



§ QUADRILATERALS AND BEYOND



This section is devoted to the smallest non-simplex. Let Q be a quadrilateral in the plane. 
Some of its moments were already explicitly shown in Example <ref>.
We know the normalized moment generating function  from Section <ref>.
The only non-faces of the quadrilateral Q are its two diagonals. 
Hence, the adjoint  Ad_Q is given by the intersection point of these diagonals. 
More specifically, if x_1, x_2, x_3, x_4 denote the cyclically labeled vertices of Q and (δ_1, δ_2) is the diagonal intersection point, then the normalized moment generating function of Q equals 

    1-δ_1 t_1-δ_2 t_2/(1-x_11t_1 - x_12t_2)(1-x_21t_1 - x_22t_2)
    (1-x_31t_1 - x_32t_2)(1-x_41t_1 - x_42t_2).


It is a non-trivial task to compute relations among the moments of quadrilaterals.
The easiest relations are given by Theorem <ref>, if we take the Hankel matrix (<ref>) for r=6.

 
Consider the moments m_i0 where i =0,1,…,6.
The corresponding moment variety
ℳ_{{ 6 }}(Q) is the
hypersurface ℳ_{{ 6 }}(2,4) ⊂^6. It is defined by
the determinant of

    ( [       0       0    m_00  3 m_10  6 m_20;       0    m_00  3 m_10  6 m_20 10 m_30;    m_00  3 m_10  6 m_20 10 m_30 15 m_40;  3 m_10  6 m_20 10 m_30 15 m_40 21 m_50;  6 m_20 10 m_30 15 m_40 21 m_50 28 m_60 ]).

This relates the moments
 of the pushforward measure given from projecting Q onto a line.


What we are actually interested in are mixed relations, i.e. equations 
in the moments m_ij  that do not come from projections onto lines as in 
Example <ref>.
The dimension of ℳ_𝒜(Q) in ℙ^|𝒜| is eight if 
𝒜⊂ℕ^2 is big enough.
We first show an interesting scenario with |𝒜| = 8.

Let 𝒜 :=( { 0,1,2 }×{ 0,1,2 })
\{ (0,0) }.
These moments are algebraically independent. Hence the moment variety
ℳ_𝒜(Q) is equal to the ambient space ^8. Consider
the map (ℂ^2)^4 ^8
which sends quadrilaterals to their moments in 𝒜.
A computation with the software HomotopyContinuation.jl <cit.>
reveals that randomly chosen fibers of this map consist of 80 points over .
We conclude that the map (ℂ^2)^4 ^8
is generically 80-to-1. The dihedral group of order 8
acts on each fiber by permuting vertices of Q.
Hence each fiber consists of 10 geometric configurations, generally over .

For a concrete example, consider the quadrilateral
X = {(1,-1), (3,2), (2,4), (-1,2)}. The fiber for this X
consists of 80 real points. 
These correspond to four non-convex quadrilaterals, two convex quadrilaterals and 
four quadrilaterals with self-crossings; see Figure <ref>.




In what follows we consider sets 𝒜 with |𝒜| = 9.
Here, the moment variety ℳ_𝒜(Q) is  typically a hypersurface in ℙ^9.
The most natural index sets 𝒜 arise from partitions
 of the integer 10. Given a partition
 λ={λ_0 ≥λ_1 ≥⋯≥λ_s > 0}, the
 corresponding index set is
 
    𝒜_λ   :=   {  (0,1), (0,2), …, (0,λ_0-1), (1,0),(1,1), …, (1,λ_1-1), …, (s,0),…, (s ,λ_s-1) }.

We simply write ℳ_λ := ℳ_𝒜_λ.
For example, ℳ_4 3 2 1(Q) is the variety of moments up to order three 
 which was  earlier denoted by ℳ_[3](Q).
 We determine this hypersurface explicitly.

 
Let Q be a quadrilateral in the plane. 
The moment variety ℳ_[3](Q) is a hypersurface in ^9, whose
defining polynomial has 5100 terms of degree ( 18,12,12). It equals

    2125764   h^6
      + 5484996  m_00^2 h^4 s
      - 1574640  m_00 g h^3
        + 364500  m_00^3h^3t
          + 3458700  m_00^4 h^2 s^2     
    
      - 2041200 m_00^3 g h s 
      + 472500 m_00^5 h s t
      -122500 m_00^6 s^3   
      +291600 m_00^2 g^2
      -135000 m_00^4 g t
      +15625 m_00^6 t^2 ,

where the affine invariants in (<ref>) are normalized
to have content one and leading monomials

    s = m_00 m_02 m_12 m_30 +  ⋯ ,   
         t = m_00^2 m_03^2 m_30^2 +  ⋯ ,  
         h = m_00 m_02 m_20 +  ⋯ ,  
         g = m_00^3 m_02^3 m_30^2 + ⋯.




The above formula was found as follows.
By Table <ref> below,   the ℤ^3-degree of the hypersurface is (18,12,12).
We used Proposition <ref> to generate affine invariants of this degree with indeterminate coefficients.
By plugging in the moments m_ij
from various random quadrilaterals, we created a system of
linear  equations in the coefficients. This system was solved which led to the formula above. Independent verification 
of the formula was carried out by checking that it vanishes on the parametrization 
(^2)^8 →ℳ_[3](Q). 


We demonstrate the same technique for another interesting
hypersurface in ^9 that is also invariant under Aff_2.
It represents the moments of order ≤ 3 of probability measures
on the triangle Δ_2 whose densities are linear functions.
This hypersurface is the image of the
8-dimensional variety ^2 ×ℳ_[4](Δ_2) 
under the map into ^9 whose coordinates are

    M_ij   =    α· m_i+1,j  + β·  m_i,j+1  + γ· m_i,j       for  
    0 ≤ i+j ≤ 3.

Here (α : β : γ ) ∈^2 and m_i,j are the moments of
the uniform probability measure on Δ_2.

 
The above hypersurface has degree ( 52,36,36). Its defining polynomial is


    12288754756878336m^16 s^9
     -125913170530271232 h^2 m^14 s^8
     -11555266180939776 h m^15 s^7 t
     -423695444226048 m^16 s^6 t^2 
    
    -242587475329941504 h^4 m^12 s^7 
    -67888179490848768 h^3 m^13 s^6 t
    -2253544388296704 h^2 m^14 s^5 t^2
    +92156256976896 h m^15 s^4 t^3 
    
    +4239929831616 m^16 s^3 t^4
    -2425179321925632 g h m^13 s^7 
    +767341894828032 g m^14 s^6 t
    -1302706722212675584 h^6 m^10 s^6 
    
    -108262506929061888 h^5 m^11 s^5 t
    +673312350928896 h^4 m^12 s^4 t^2 
    +535497484271616 h^3 m^13 s^3 t^3 
    +31959518257152 h^2 m^14 s^2 t^4 
    
    +440798423040 h m^15 s t^5
    +195936798885543936 g h^3 m^11 s^6
    -410140620619776 g h^2 m^12 s^5 t 
    -412398826108747776 g h^6 m^8 s^3 t 
    
     -2360537593675776 g h m^13 s^4 t^2
    -89805332054016 g m^14 s^3 t^3
    -486870353365172224 h^8 m^8 s^5
    +6819936693387264 h^7 m^9 s^4 t 
    
    +29422733985054720 h^6 m^10 s^3 t^2 
    +2782917213290496 h^5 m^11 s^2 t^3 
    +58246341746688 h^4 m^12 s t^4 
    -587731230720 h^3 m^13 t^5  
    
    +3602104581095424 g^2 m^12 s^6 
    -157746980481662976 g h^5 m^9 s^5 
    -79828890012352512 g h^4 m^10 s^4 t
    -10700934975848448 g h^3 m^11 s^3 t^2 
    
    -668738492301312 g h^2 m^12 s^2 t^3
    -10448555212800 g h m^13 s t^4
    +275499014400 g m^14 t^5 
    +1321196639636946944 h^10 m^6 s^4 
    
    +814698134331457536 h^9 m^7 s^3 t
    +92179893357379584 h^8 m^8 s^2 t^2
    +2541749079638016 h^7 m^9 s t^3
    -13792092880896 h^6 m^10 t^4 
    
    +58678654946770944 g^2 h^2 m^10 s^5
    +16167862146170880 g^2 h m^11 s^4 t
    +705486447968256 g^2 m^12 s^3 t^2
    -1103687847816200192 g h^7 m^7 s^4 
    
    +13931406950400 g h^3 m^11 t^4 
    -44584171418419200 g h^5 m^9 s^2 t^2 
    -9685512225 m^16 t^6
    -1132386035171328 g h^4 m^10 s t^3  
    
    +7839053087502237696 h^12 m^4 s^3 
    +1352219532013338624 h^11 m^5 s^2 t 
    +51427969540816896 h^10 m^6 s t^2
    -147941222252544 h^9 m^7 t^3 
    
    +356552602772570112 g^2 h^4 m^8 s^4
    +65355404946702336 g^2 h^3 m^9 s^3 t
    +5201278745444352 g^2 h^2 m^10 s^2 t^2 
    +99067782758400 g^2 h m^11 s t^3 
    
    -3265173504000 g^2 m^12 t^4
    -5301992678571900928 g h^9 m^5 s^3
    -984505782412247040 g h^8 m^6 s^2 t
    -37440870596739072 g h^7 m^7 s t^2 
    
    +260713381625856 g h^6 m^8 t^3
    +7163309458867617792 h^14 m^2 s^2
    +495888540219998208 h^13 m^3 s t
    -613682107121664 h^12 m^4 t^2 
    
    -33414364526542848 g^3 h m^9 s^4 
    -2441030167166976 g^3 m^10 s^3 t
    +1297818789047435264 g^2 h^6 m^6 s^3
    +235088951956733952 g^2 h^5 m^7 s^2 t 
    
    +8250658482290688 g^2 h^4 m^8 s t^2
    -132090377011200 g^2 h^3 m^9 t^3 
    -7123133303988682752 g h^11 m^3 s^2
    -506754841838616576 g h^10 m^4 s t 
    
    +2079004689432576 g h^9 m^5 t^2
    +1846757322198614016 h^16 s
    -126388861612851200 g^3 h^3 m^7 s^3 
    -17847573389770752 g^3 h^2 m^8 s^2 t 
    
    -469654673817600 g^3 h m^9 s t^2
    +20639121408000 g^3 m^10 t^3
    +2594242435278176256 g^2 h^8 m^4 s^2
    +183620365983940608 g^2 h^7 m^5 s t 
    
    -1848091141472256 g^2 h^6 m^6 t^2
    -2445243491429646336 g h^13 m s
    +5610807836540928 g h^12 m^2 t
    +3143555283419136 g^4 m^8 s^3 
    
    -408993036765233152 g^3 h^5 m^5 s^2 
    -26702361435045888 g^3 h^4 m^6 s t
    +626206231756800 g^3 h^3 m^7 t^2
    +1246806603479384064 g^2 h^10 m^2 s 
    
    -9737274975584256 g^2 h^9 m^3 t 
    +22822562857746432 g^4 h^2 m^6 s^2 
    +1113255523123200 g^4 h m^7 s t
    -73383542784000 g^4 m^8 t^2 
    
    -299841218941026304 g^3 h^7 m^3 s 
    +5822326385934336 g^3 h^6 m^4 t
    -12824703626379264 g^2 h^12 
    +32389413531025408 g^4 h^4 m^4 s 
    
    -1484340697497600 g^4 h^3 m^5 t 
    +15199648742375424 g^3 h^9 m
    -1055531162664960 g^5 h m^5 s
    +139156940390400 g^5 m^6 t 
    
    -6878544743366656 g^4 h^6 m^2
    +1407374883553280 g^5 h^3 m^3
    -109951162777600 g^6 m^4.

Here m= m_00 and s,t,h,g are the
affine invariants in Example <ref>  and
 Theorem <ref>.


We now return to the hypersurfaces ℳ_λ(Q) 
that encode moments
of the uniform probability distribution on a quadrilateral Q.
These also live in ^9 but they
are not invariant under Aff_3.
We consider arbitrary partitions λ of 10 and notice that 
their total number  is 42.


 
For every partition λ of 10, except those in the following table, the moment variety ℳ_λ(Q) is a hypersurface in ℙ^9.
The dimensions of the remaining moment varieties coming from partitions of 10 are as follows.
Here λ^c denotes the conjugate partition of λ.



  2
  λ     λ^c     ℳ_λ(Q) 

  10     1^10     5 

 9 1     2 1^8     6 

 8 2     2^2 1^6     7 

 8 1^2     3 1^7     7
  

  2



 

In light of Theorem <ref>, we find that all equations for moment varieties in this table arise from projections onto a line.
In particular, adding either m_10,m_11,m_12 or m_10,m_11,m_20 or m_10,m_20,m_30 to the moments m_00,m_01,…,m_06 does not impose any new relations.
The hypersurfaces ℳ_7 3, ℳ_7 2 1 and ℳ_7 1^3 are all cut out by the same Hankel determinant (<ref>).


We now come to the census of mixed relations we are interested in.
These are the moment hypersurfaces ℳ_λ(Q) in ^9 that are not featured in
Remark <ref>. One of them is defined by the polynomial
of degree 18 seen in Theorem <ref>. The other hypersurfaces
are not invariant under Aff_3. We computed all of them
using numerical algebraic geometry. Here is the result:

 
Table <ref> lists the ℤ^3-degrees of the moment hypersurfaces ℳ_λ(Q) in ℙ^9, where
Q is a quadrilateral and λ is a partition of 10. 
We also report  the size of the general fiber of the map
φ_λ:(ℂ^2)^4 →ℳ_λ(Q)
which sends the vertices of Q to the moments indexed by λ. 





This is based on numerical computations. We started out with
 <cit.>, but then we mainly used the 
  package  <cit.>.
 
Consider the parametrization of the affine cone over the moment hypersurface ℳ_λ(Q) given by ℂ^9 →ℂ^10, (t,X) ↦ t ·φ_λ(X).
Let us first describe how we compute the usual degree of this affine cone in ^10.
We pick a random point on the  cone together with a random line passing through this point. Our goal is to compute all intersection points of the line with the cone. 
We do this via numerical monodromy, i.e. we move the line around and track the already known intersection point. 
When the original line is reached again, we might have found a new solution.
These monodromy loops are executed until no new solutions are found. 
To verify that all solutions have been found, we applied the trace test <cit.>.

To compute the other two coordinates
in the ℤ^3-degree of the moment hypersurface ℳ_λ(Q), we proceed as above, but the line is now replaced by a monomial curve. For the middle coordinate of
the ^3-degree, we use the curve in ^10 with parametric representation

    s ↦( p_1 + s^i_1 v_1,   p_2 + s^i_2 v_2,  …,   p_10 + s^i_10 v_10).

Here p and v are random vectors in ℂ^10.
 The moments indexed by the partition λ appear in the 
 order m_i_1,j_1, m_i_2,j_2, …, m_i_10,j_10.
Analogously, for the last entry in the ℤ^3-degree, we use the
monomial curve in ^10 parametrized by 
s ↦( p_1 + s^j_1 v_1,    p_2 + s^j_2 v_2,  …, p_10 + s^j_10 v_10).

In each case, we solve a square system of 10 polynomial
equations in 10 unknowns s,t,x_11,…,x_42.
The number of solutions is the desired degree in ^10
times the degree of the map φ_λ. For instance,
the number of solutions (s,t,x_11,…,x_42)
 for λ = (4,3,2,1) equals 144.
The solutions  form 18 clusters of size 8, where each cluster
consists of all solutions   that map to the
same point on the affine cone. This is how the degree 18
was first determined. It allowed us to make the ansatz 
that eventually led to the invariant in Theorem <ref>.


The use of invariant theory of the affine group Aff_d
was essential for computing the moment hypersurfaces 
in Theorem <ref> and Proposition <ref>.
However this method does not directly apply to moment varieties of codimension two or more.
For such moment varieties, the minimal generators of the ideal
form an invariant vector space, but the individual generators
are not invariants. In such a situation, one might employ
 representation theory of Aff_d. We shall demonstrate this for the 
 moment variety ℳ_[3](Δ_3) in Conjecture <ref>.

 
The Aff_3-module V spanned by the 90 quintics 
that vanish on ℳ_[3](Δ_3) in ^19
is the direct sum of two indecomposable Aff_3-modules
 V_1 and V_2, each of dimension 45.
As a GL_3-module, V decomposes into 12 irreducibles:
V_1 and V_2 split into six irreducible GL_3-modules each. 
Table <ref> lists
the highest weights of these GL_3-modules and their dimensions.





The weight of a polynomial is given by its ℤ^4-grading.
Each isotypical component of V as a GL_3-module is spanned by all polynomials in V having the same fixed ℤ^4-degree.
This isotypical decomposition consists of 43 vector spaces with dimensions 1, 2, 4 or 6.

For each isotypical component, we computed its U_3-invariant polynomials, where U_3 ⊂GL_3 is the subgroup of upper triangular matrices with diagonal (1,1,1).
Ten isotypical components contain exactly one U_3-invariant (up to scaling),
while the component with weight (2,3,4) has a two-dimensional subspace of U_3-invariant polynomials; see Table <ref>.
Each  U_3-invariant  generates an irreducible GL_3-module.
Ten of these irreducible modules in V are unique. 
The two irreducible GL_3-modules with highest weight (2,3,4) are not unique.

Finally, we studied which of the described irreducible GL_3-modules merge when we add translation, i.e. when we act on V by the whole affine group Aff_3.
The ten unique GL_3-modules get merged into two clusters, as seen in Table <ref>.
Moreover, there is a unique way of choosing two GL_3-modules with highest weight (2,3,4) such that acting with the affine group on one of these modules stays within one of the two clusters in Table <ref>.




§ OUTLOOK



Moment varieties furnish an algebro-geometric representation for 
various probability measures on ^d. In this
article we focused on measures that are associated with convex polytopes.
We were able to determine their moment varieties for a range
of interesting cases. However, this is just the beginning.
Many questions remain open, and we see considerable
potential for further developing our algebraic tools,
so that they become  practical for inverse problems.

This section discusses a number of open problems and directions
for future research. It also offers a perspective on some
aspects of moment varieties not discussed in Sections <ref>–<ref>.

Adjoints and Wachspress Varieties. 









At the end of Section <ref> we defined the adjoint moment 
variety ℳ_ Ad() for a given combinatorial type , 
but we did not state any results on this topic.
The variety ℳ_ Ad() is the moduli space
for the Wachspress varieties of the polytopes  in the class .
The study of Wachspress varieties and their moduli
is a promising direction at the interface of geometric
combinatorics  and algebraic geometry (see <cit.>). It extends the familiar repertoire
of toric varieties.

A concrete open problem is to compute the
adjoint moment variety ℳ_ Ad()
in the smallest cases where 
the ambient dimension n-1d-1
exceeds the number nd of parameters.
This happens for polytopes 
with n=8 vertices
in dimensions d=2,3,4.
Another interesting case is d=2 and n=7.
Here the adjoint is a plane curve of degree 4,
so it has 14 parameters. It is parametrized
by the 14 vertex coordinates of a heptagon.
What is the degree of this map?  It would be
worthwhile to study  the geometry of this map,
in light of the beautiful classical
connections  <cit.>
between genus 3 curves and
del Pezzo surfaces of degree 2.

Step Functions.
  It can be shown that mixtures
 of uniform distributions of line segments are
 algebraically identifiable whenever this is permitted
 by the parameter count.
 To be precise, the delicate algebro-geometric proof for
  mixtures of univariate Gaussians that is
 given in <cit.> can be transferred to mixtures of
 line segments. The point of departure for this transfer argument is the proof of
<cit.> which holds verbatim for the matrix in (<ref>).

This opens the door to moment varieties of distributions whose
density is a  step function on the line ^1.
Indeed, each step function is a mixture of uniform distributions on
line segments. Since mixture models correspond to secant varieties in ^r,
we can phrase our question as follows: study the secant varieties of the
surfaces ℳ_{{ r }}(1,2) in Example <ref>.
Pearson's hypersurface of degree 39 in <cit.>
suggests that this will not be easy.

Recovery Algorithms.  Theorem <ref> characterizes all relations among axial moments  of a polytope P for any fixed axis. From this one can recover
  the projections of all vertices of P onto that axis.
  Different variations of this result are known in the literature; see e.g. <cit.>. 
  On the other hand, in order to uniquely recover  a polytope P in ^d  using 
  axial  moments, one has  to know the projections of its vertices on at least d+1 different lines in ^d. The moments on d+1 lines are highly dependent.
  For instance, for d=2 and P a quadrilateral, the  λ = 6   1^4 entry 
  in Table <ref> reveals a relation of degree 480 among moments on two axes.
Understanding such dependencies among the axial moments for general polytopes
seems difficult, but it is an important step towards developing more
advanced recovery algorithms.
This issue is related to multidimensional variants of Prony's method.
Indeed, the Hankel matrix (<ref>) which connects
polytopal densities and its node points on ^1 
with the axial moments is analogous to that for  the classical Prony system <cit.>.
Extending known results about the Prony system to our setting
in ^d may lead to applications in signal processing.

Multisymmetric Functions.
Let [X] denote the ring of polynomials in the entries
of an n × d matrix of unknowns X = (x_kl). The symmetric group S_n
acts on [X] by permuting the rows of X.
Following Dalbec <cit.>, we write Λ_d,n = [X]^S_n
for the ring of invariants under this action. In words, Λ_d,n is the
ring of multisymmetric functions for n vectors in d-space.

The case n=d+1 appeared in Section <ref>. 
 Proposition <ref> and Corollary <ref> imply
 that the moments of simplices in ^d generate the ring Λ_d,d+1.
 Indeed, the moments and the cumulants generate the same algebra, and
 the  cumulants coincide with the power sum multisymmetric polynomials.
By <cit.>, the latter are known to generate Λ_d,n for any n.
Furthermore, our Proposition <ref> is closely related to the
well-known fact  (cf. <cit.>)
that elementary multisymmetric polynomials also generate the algebra Λ_d,n.
  
The discussion at the end of Section <ref> shows that, for any n > d,
the ring Λ_d,n arises from our polytopal measures.
Namely, consider the projection of an (n-1)-simplex to a subspace ^d.
Suppose that  the image is a d-polytope with n vertices.
The moments of the induced polytopal measure are multisymmetric polynomials in
Λ_d,n, and, in fact, these moments generate the invariant ring Λ_d,n.
Therefore we obtain all possible rings of multisymmetric polynomials as  special cases of the rings of moments of simplices and their projections.  It is known in algebraic combinatorics
that these rings are quite complicated, see e.g. <cit.>. 

Symmetry and Invariants.  We demonstrated in Section <ref> that
invariants of the affine group can be determined from covariants of the
general linear group, and this was used in Section <ref> to give explicit
formulas for two specific moment hypersurfaces in ^9. In the case
of moment varieties of codimension ≥ 2, we do not really know 
how to take advantage of symmetries arising from the affine group Aff_d.
It would be desirable to understand this.

More Hypersurfaces. 
In Theorem <ref> we determined many
moment hypersurfaces of quadrilaterals in ^9, one for each
partition λ of the integer 10. Our computations were based
on methods from numerical algebraic geometry. One could
try to push this further, either to pentagons (d=2,n=5)
or to tetrahedra (d=3,n=4). In the former case
we would aim for moment hypersurfaces in ^11 associated
with partitions of 12, and in the
latter case we would seek  moment hypersurfaces in ^13
associated with plane partitions of 14. The remark after
Proposition  <ref> suggests the following problem
for numerical algebraic geometry:
compute the degrees of the moment hypersurfaces
ℳ_[6](13-gon) ⊂^27 
and ℳ_[3]( octahedron )⊂^19.
 
Special Subvarieties.
It would be interesting to study the singular loci of moment varieties
as well as the subvarieties whose points correspond to degenerate geometric configurations. 
This was discussed for the cubic surface in Figure <ref> but we never returned to that topic.
 
Moment Rings of Polytopes.  Fix a combinatorial type 𝒫 of
simplicial polytopes in ^d. We define  the moment ring _
to be the subalgebra of the rational function field (X) that is generated
by the moments m_I(X) for 𝒫 where I runs over  ^d.
We can realize _ as the subalgebra of the polynomial ring [X],
 generated by the numerators m_I(X) · vol(X).
These products are polynomials in the nd unknowns x_kl 
by Theorem <ref>. 
If P is the d-simplex  then the moment ring _
is the ring Λ_d,d+1 of multisymmetric polynomials, as discussed above.
A priori, it is not even clear that _ is a Noetherian ring. However,
we strongly believe this. In other words, we conjecture
that _ is finitely generated. It would be very interesting to
identify explicit generators, or, at least, to find
degree bounds for the generators of _.
The same question makes sense for the moment rings
 that are analogous to Λ_d,n for n > d+1.
To be specific, we seek the subalgebra of (X) that is
generated by  all moments  of univariate polytopal measures of type (d,n). 
A natural place to start is the case of the convex n-gon in the plane.
Here we might take advantage of the dihedral group acting
on the n vertices.
In the case of the ring generated by all harmonic moments of plane polygons such study was carried out in <cit.>.

The group of symmetries of the combinatorial type  acts on its moment ring 
_. This explains why the moment ring of a simplex consists of multisymmetric functions and why the dihedral group acts on the moment rings of n-gons. Of course,
there are many other types of simplicial polytopes with interesting symmetry groups.
  How about the octahedron?






  
Acknowledgments.
We thank Jan Draisma, Frank Grosshans and Hanspeter Kraft  for 
communications on invariant theory.
We are grateful to Taylor Brysiewicz, Paul Breiding and Sascha Timme for
helping us with our experiments using numerical algebraic geometry.
Kathlén Kohn and Boris Shapiro are grateful to the MPI MIS in Leipzig 
for the hospitality in June 2018 where this project was initiated.
Bernd Sturmfels also acknowledges partial support
from the Einstein Foundation Berlin and the US National Science Foundation.





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Authors' addresses:



Kathlén Kohn,
ICERM, Brown University
kathlen.korn@gmail.com

Boris Shapiro, Stockholm University
shapiro@math.su.se

Bernd Sturmfels,
   MPI-MiS Leipzig and
UC  Berkeley bernd@mis.mpg.de

