Signatures of partition functions and their complexity reduction through the KP II equation
Mario Angelelli
math-ph math.MP


Signatures of partition functions and their complexity reduction]Signatures of partition functions and their complexity reduction
through the KP II equation

[
    Balajewicz M.
    December 30, 2023  File: 
=============================







Department of Mathematics and Physics 
“Ennio De Giorgi”, University of Salento and sezione INFN, 
Lecce 73100, Italy 




A statistical amoeba arises from a real-valued partition function
when the positivity condition for pre-exponential terms is relaxed,
and families of signatures are taken into account. This notion lets
us explore special types of constraints when we focus on those signatures
that preserve particular properties. Specifically, we look at sums
of determinantal type, and main attention is paid to a distinguished
class of soliton solutions of the Kadomtsev-Petviashvili (KP) II equation.
A characterization of the signatures preserving the determinantal
form, as well as the signatures compatible with the KP II equation,
is provided: both of them are reduced to choices of signs for columns
and rows of a coefficient matrix, and they satisfy the whole KP hierarchy.
Interpretations in term of information-theoretic properties, geometric
characteristics, and the relation with tropical limits are discussed. 




Keywords: Statistical amoeba, soliton, determinant, complexity.

MSC: 15A15, 35C08, 82B05



§  INTRODUCTION


The concept of partition function encodes in a single object the statistical
data compatible with the physical constraints of a system, e.g., conservation
laws. Hence, it explicitly connects its probabilistic and physical
characteristics. This is a fundamental principle in the investigation
of composite systems <cit.> and is now applied
in many different branches of sciences <cit.>. The
basic form of the partition function involves a sum of exponential
terms 

    𝒵:=∑_α∈𝕀g_α· e^-ε_α/k_BT

where k_B is Boltzmann's constant, T is the temperature, ε_α
are the energy levels labelled by an indexing set 𝕀 and
g_α are the associated degenerations.

In practice, the implementation of this approach is limited by the
intrinsic complexity of the model, since there are few cases where
the partition function can be evaluated exactly, e.g., in a closed
form <cit.>. Indeed, many phenomena in complex systems
cannot be reduced to their individual components, thus (<ref>) involves
collections of objects and, consequently, the complexity of the calculation grows exponentially with the size of the system.

The occurrence of exact formulas for the partition function implies
relations between the characterizing quantities of the system, e.g.,
correlation functions that are generated by the partition function
through derivation. This gives rise to a family of differential equations
whose compatibility follows from the existence of the partition function.
On the other hand, one can start from a family of differential equations
and look for their compatibility. This naturally leads to the investigation
on connections between the partition function formalism and the concept
of integrability, especially integrable hierarchies. In such a context,
the role of the partition function is similar to the notion of τ-function,
which provides one with a unifying framework for hierarchies of nonlinear
partial differential equations and their remarkable behaviours, e.g.,
symmetries, infinite family of commuting flows, soliton solutions
<cit.>.

Explicit links between the partition function formalism and integrable
systems have already been identified and used to provide fundamental
techniques in modern theoretical physics <cit.>. Remarkably,
certain solutions of integrable PDEs can be interpreted as potentials,
such as the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations in
topological quantum field theories. A solution for the WDVV system
defines a free energy for this theory, namely, a function which generates
correlators by means of derivation <cit.>.

Recent studies have been devoted to the combinatorics underlying these
kinds of structures. One of most fruitful examples is the Kadomtsev-Petviashvili
II (KP II) equation 

    ∂_x(-4·∂_tu+∂_xxxu+6u·∂_xu)=-3∂_yyu

where ∂_xu denotes the partial derivative of u≡ u(x,y,t)
with respect to x. The KP II equation is one of most important
(2+1)-integrable PDEs and is considered a universal model of two-dimensional
integrable evolutionary equations. The combinatorial structures arise,
for example, taking into account particular classes of solutions of
the KP II equation, such as soliton solutions of determinantal (Wronskian)
type <cit.>. These solutions are parametrized by points
in the real Grassmannian space and their regularity and algebraic
features have been extensively studied in the last years <cit.>.

The aim of this work is to explore these requirements starting from
a statistical perspective: we choose a class of combinatorial configurations
related to sums of exponentials that generalise (<ref>).
Then, we investigate the information content encoded in constraints
through the reduction of the class of allowed configurations. This
purpose will be realised extending the method of statistical amoebas
introduced in <cit.>, i.e., considering a family of discrete
“deformations” of the partition function and focusing on those
that are compatible with some given constraints. A statistical amoeba
is obtained relaxing the assumption of positive degenerations g_α
in (<ref>). Negative degenerations
of energy levels can be related to an imaginary part of energies ε_α
and, hence, used to describe metastable states <cit.>.
This approach has proved useful in statistical physics, where the
locus of zeros of complex-valued partition functions is employed in
the analysis of phase transitions <cit.>.
In our context, the partition function is real-valued, but indefinite
signs for degenerations g_α open up the way to the study
of stability (𝒵>0), instability (𝒵<0) and
phase transitions (𝒵=0).

In general, the requirement of compatibility of the choices of signs
for g_α with a given constraint affects both the number
and the form of allowed configurations. We will focus on determinantal
relations, where the complexity of calculations of determinants is
polynomial via Gaussian reduction (while other immanents, in general,
have exponential complexity <cit.>), and integrability,
which reduces the complexity through the occurrence of conserved quantities.
We stress this point referring to the construction in <cit.>,
where all the combinations of signs for the N exponential terms
in the partition function are allowed, as a free statistical
amoeba. On the other hand, restrictions on the allowed combinations
of signs give rise to a constrained statistical amoeba.

A strong relation between determinantal relations and integrability
has been established by Sato (see, e.g., <cit.> and further
developments in <cit.>), where an expansion in terms of
Schur functions is found to be a τ-function for the KP hierarchy
if and only if its coefficients satisfy the Grassmann-Plücker
relations <cit.>. This has led to intensive
research on equivalent reformulations of the KP hierarchy such as
these involving the infinite Grassmannian <cit.>.
Whilst sharing some concepts with these issues, the present investigation
is devoted to families of functions generated by a special class of
solutions of the KP equation. It should be remarked that
we concentrate only on the first equation (<ref>)
in the hierarchy for what concerns constraints reducing the complexity,
and that we work in finite dimensionality. Moreover, the construction
of statistical amoebas does not rely only on the global (determinantal)
form of the initial function, but also on the individual terms
in its expansion, because the generation of other functions goes through
changes of signs for a given expression of the type (<ref>).
The consistency check with constraints and the construction of allowed
configurations depend on relations between non-vanishing terms.

When we start from solitons in Wronskian form, the constraints come
either from the determinantal relations or from the fulfilment of
the KP II equation. In fact, we will see that both these assumptions
reduce the allowed combinations of signs to the same family, i.e.,
the set of transformations that can be obtained by flipping the signs of some rows and/or columns of a coefficient matrix 𝐀:
this is showed in Theorem <ref>.
The complexity reduction takes place because the information about
the chosen signature for the terms in the exponential sum, which are
labelled by subsets of {1,…,n}, is stored in the elements
of {0,1,…,n}: in general, this presentation is non-unique
due to a certain equivalence relation (Proposition <ref>).
The connection with other equations in the hierarchy follows from
the special form of the transformations in this family (Theorem 
<ref>).

The main scope of this work is the formal derivation of the above-mentioned
properties. However, it is important to highlight potential applications
of this approach: for instance, choosing a coefficient matrix and
constructing the corresponding soliton solution, one can encode a
sequence of “bits” in the signs of the pre-exponential terms,
and recover a special family of sequences checking that the KP II
equation is satisfied. We will quantify the information content following
from the check of the KP II equation through the Kullback-Leibler
divergence between two probability distributions on the strings of
signatures. These results also have geometric implications, and we
will briefly discuss the links with oblique projections and their
statistical relevance. Likewise, we will point out the connection
with the tropical limit in statistical physics as introduced in <cit.>
and developed in <cit.>. These issues could be of
interest for a better understanding of the relation between statistical
physics, complex systems and learning methods <cit.>.
We will outline some of these connections and postpone their detailed
study to a separate article.

The paper is organized as follows: in Section <ref>
we fix the notation and summarize the basic notions that provide a
starting point for our investigation. In Section <ref>
we introduce the link between soliton solutions of the KP hierarchy
and statistical amoebas, generalizing the latter to higher dimensions.
The occurrence of a particular structure to be preserved in the construction
of statistical amoebas is the focus of the next sections: in Section
<ref> we prove that
the compatibility of a choice of signs for the coefficients of exponential
terms with the KP II equation implies that it is induced by rows and
columns sign flips for a coefficient matrix. In Section <ref>
we consider the number of distinct configurations that can be obtained
in this way. The relation with the strata of the corresponding free
amoeba <cit.> is considered in Section <ref>. Two applications of the present framework are addressed in Section <ref>: the information content in the KP II constraint is studied via the Kullback-Leibler divergence (Subsection <ref>), while an intersection property is  discussed in geometric terms (Subsection <ref>). Finally we draw conclusions and address some issues of potential interest for future investigations in Section <ref>.



§  PRELIMINARIES


We briefly summarise the basic state of the art regarding Wronskian
soliton solutions of the KP II equation and the statistical amoeba
formalism which will be used in the rest of the paper. Before that,
it is worth introducing some notations to enhance clearness.



 §.§  Notation 


In the following, we will denote by 𝒫[n] the power set
of [n]:={1,…,n}, by 𝒫_k[n] the collection
of subsets of [n] with k elements and by ℐΔ𝒥:=(ℐ∖𝒥)∪(𝒥∖ℐ)
the symmetric difference of ℐ,𝒥∈𝒫[n].
We will also use the notation {±1}:={+1,-1}, 

    ℐ_β^α:=ℐ\{α}∪{β}  α∈ℐ,β∉ℐ

and, similarly, ℐ_β:=ℐ∪{β},
ℐ^α:=ℐ\{α}, ℐ^α_1α_2:=ℐ\{α_1,α_2},
etc. The expression ℐ_β^α implicitly assumes that α∈ℐ, and β∉ℐ, unless α=β, in which case we have ℐ_α^α:=ℐ.

The symbol Δ_𝐀(ℐ) (respectively Δ_𝐊(ℐ))
is the maximal minor of 𝐀∈ℝ^k× n (respectively,
𝐊) whose columns (respectively, rows) are indexed by ℐ∈𝒫_k[n].
We will occasionally use Δ(𝐀;ℐ) instead
of Δ_𝐀(ℐ) for the sake of clearness.
When a permutation π of [n] is involved, an additional sign
comes from the parity of the number of inversions induced by π
on ℋ. Particular attention will be paid to the set of
pivot columns 𝒱:={ν_1,…,ν_k}, which is
the least element of 𝒫_k[n] (with respect to the lexicographical
order) associated with a non-vanishing minor Δ_𝐀(𝒱).

The (n× k)-Vandermonde
matrix relative to real parameters κ_1,…,κ_n
is 𝐊:=(κ_α^i-1)_α∈[n]^i∈[k]. The determinant of a general (not necessarily maximal) minor of 𝐊 is 

    VdM(;ℋ):=(𝐊|_ℋ×[k])=∏_α<β^α,β∈ℋ(κ_β-κ_α),  ℋ⊆[n].

So 𝐊 has maximal rank in cases of pairwise distinct soliton
parameters κ_1,…,κ_n, as we will assume.
We will look at exponential sums whose terms also depend on variables
x∈ℝ^d, d≥3, where x_1=x, x_2=y,
and x_3=t.

The real Grassmannian 𝒢ℛ_k,n(ℝ) is the space
of k-dimensional linear subspaces of a n-dimensional vector
space over ℝ. It can be presented as the quotient of the
space of k-dimensional frames in ℝ^n by the left
action of GL_k(ℝ). If ℝ_⋆^k× n
denotes the set of real (k× n)-matrices of maximal rank min{k,n}=k,
one gets 

    𝒢ℛ_k,n(ℝ)≅ℝ_⋆^k× n/GL_k(ℝ)

where GL_k(ℝ) acts as left multiplication. Greek indices
α,β∈[n] will often represent columns, while Latin indices
i,j∈[k] will be used to label rows. Additional notation will
be introduced in specific paragraphs.



 §.§ 
KP II equation and Wronskian solutions


Determinantal solitons define a distinguished class of solutions that
can be derived from Hirota's direct method <cit.> introducing derivatives D_x acting on pairs of functions: 

    D_x(f· g):=(∂_xf)· g-f·(∂_xg)=(∂_x_1-∂_x_2)f(x_1)g(x_2)|_x_1=x_2.

So one can rewrite the KP II equation (<ref>)
in the following homogeneous bilinear form 

    D_KP(τ,τ):=(D_x^4-4· D_xD_t+3· D_y^2)τ·τ=0.

By the same token, one can rewrite the other equations of the KP hierarchy
in bilinear form <cit.>. The tau-function τ(x)
of the KP equation is related to u via 

    u(x):=2·∂^2/∂ x_1^2lnτ(x).

It has been shown (see, e.g., <cit.>) that the Hirota
derivative (<ref>) can be characterized as
a derivative operator with gauge invariance under the simultaneous
action f↦ e^kxf and g↦ e^kxg, i.e., 

    D_KP(e^-ϑ(x)· f(x),e^-ϑ(x)· g(x))=e^-2ϑ(x)·D_KP(f(x),g(x))

for any linear function ϑ(x) of x.
This is manifest in the antisymmetric form of D_x in (<ref>)
and is also reflected in the expression of a special class of solutions
<cit.> 

    Wr(f_1,…,f_n)=(∂_x^(i-1)f_m)_i,m∈[k]

where f_α, α∈[n], are independent solutions for
the following system of partial differential equations 

    ∂/∂ x_rf=∂^r/∂ x_1^rf,   r∈[d].

One can take a certain number, say M, of solutions of (<ref>) in the
form 

    E_α(x)=expφ_α(x):=exp(∑_r=1^dκ_α^rx_r)

with real parameters κ_α. A particular choice of solutions
of (<ref>) comes from linear combinations
of these exponentials, with coefficients given by the entries of a
matrix 𝐀: f_i(x)=∑_α=1^nA_iαE_α(x),
that is f=𝐀·E. One
has 

    ([          E_1       ∂_xE_1            … ∂_x^(k-1)E_1;            ⋮            ⋮            ⋱            ⋮;          E_n       ∂_xE_n            … ∂_x^(k-1)E_n ])=([        E_1     κ_1E_1          … κ_1^k-1E_1;          ⋮          ⋮          ⋱          ⋮;        E_n     κ_nE_n          … κ_n^k-1E_n ])=·𝐊

where :=diag(E_1,E_2,…,E_n).
Finally, the resulting soliton solution is equivalently expressed
using the Cauchy-Binet expansion <cit.> as 

    τ(x)     :=    (𝐀·(x)·𝐊)
        =    ∑_ℐ∈𝒫_k[n]Δ_𝐀(ℐ)·Δ_𝐊(ℐ)· e^∑_α∈ℐφ_α(x).

It should be remarked that the left action of GL_k(ℝ)
on ℝ_⋆^k× n, i.e., the multiplication of 𝐀
by a full-rank k× k real matrix, induces the multiplication
of τ by a constant and, hence, leaves solutions (<ref>)
invariant. Such an action is equivalent to row operations on 𝐀,
thus soliton solutions are parametrized by points of the real Grassmannian
𝒢ℛ_k,n(ℝ) <cit.> rather than
points on ℝ_⋆^k× n (see, e.g., <cit.>).

The solution (<ref>) is regular at τ(x)>0.
If τ(x)<0, then lnτ(x) is multivalued,
but its imaginary part does not depend on x and disappears
after derivation in (<ref>). Thus, possible
singularities of soliton solutions are related to the points where
τ vanishes. If the order κ_1<…<κ_n for
soliton parameters is fixed, then the locus of zeros of τ is
not empty if there exist maximal minors with opposite sign, i.e., if
𝐀 parametrises a point outside the totally non-negative
part of the Grassmannian <cit.>.



 §.§  Statistical amoebas 


The analysis of roots of the partition function is a fundamental
technique in the study of stability, metastability and phase transitions in composite
systems <cit.>. Finite
sums of exponentials of the type (<ref>)
are positive for real values of energies E_n and temperature
T and positive degeneracies g_n. So the zeros of the partition
function define a singular locus in the complex
domain and, in many cases, they approach the real line when the number
of terms involved in (<ref>) becomes
large (thermodynamic limit).

If one restricts to the real domain, the partition function can vanish
if not all the degeneracies g_n have the same sign. This corresponds
to real partition function of indefinite signatures and relates to the concept of negative probabilities <cit.>. This proposal has been developed in <cit.> for partition functions of the type 

    𝒵:=∑_α=1^Ng_α· e^f_α(x)

where the functions f_α(x) represent “micro-free
energies” that depend on certain parameters x∈ℝ^d
(e.g., temperature, external magnetic fields, etc.). When one
fixes s≤N/2, the s-stratum of the statistical
amoeba consists of the zero loci of the functions produced by any
possible combination of s signs among the N terms in (<ref>),
i.e., g∈{±1}^N with #(g^-1({-1}))=s.
In great generality, that is under the only assumption of polynomial
functions f_α(x), α∈[N], a pattern can
be found in the study of the singular locus: the s-stratum is confined
in a region ℝ^d∖𝒟_s- of the space
of parameters ℝ^d, and the instability domains 𝒟_s- obey the following inclusion property 

    𝒟_s-⊆𝒟_ŝ-,  1≤ s<ŝ<N/2.

The restriction s<N/2 also avoids the redundancy given
by the simultaneous reversal of all the signs. This is equivalently expressed
via the involution g^-1({-1})↦[n]∖g^-1({-1}), which preserves the singular locus and exchanges the role of
equilibrium (𝒵>0) and non-equilibrium (𝒵<0)
regions, since 𝒵(-g)=-𝒵(g). Thus, the strata associated with s>N/2 are said to generate the statistical antiamoeba.

When all the Ns combinations of s
negative coefficients g_α are taken into account, the set 𝒟_s- coincides with the locus of points where the maximal number of negative partition functions (<ref>) is obtained. This maximum
is the same for all the systems with polynomial f_α and
equals N-1s-1. Furthermore, the polynomial assumption
is also suitable for the study of the tropical limit <cit.>,
both in the linear and in the nonlinear cases (the latter is referred
as a multi-scaling tropical limit <cit.>).



§  FROM SOLITON SOLUTIONS TO
STATISTICAL AMOEBAS





The explicit form of many soliton solutions of partial
differential equations can be derived from a sum of exponentials <cit.>.
In order to highlight the relation between statistical amoebas and
τ-functions, it is worth noting that the partition function
(<ref>) can be expressed as 

    𝒵=(g^T·()·K_0)

where g:=(g_1,g_2,…,g_N)^T, ():=diag(e^-ε_1/k_BT,e^-ε_2/k_BT,…,e^-ε_N/k_BT)
and K_0:=(N1,…,1)^T.
In particular, g is a totally positive vector (all its
entries are positive) and K_0 can be interpreted as
a n×1 Vandermonde matrix. This formula for the partition function
coincides with a Wronskian τ-function (<ref>).

More generally, we can express such a type of τ-functions as a sum of exponentials through the Cauchy-Binet expansion of the determinant of a product: if one introduces

    g_ℐ    :=    Δ_𝐀(ℐ)·Δ_𝐊(ℐ),
    Λ_ℐ(x)     :=    Δ_𝐀(ℐ)·Δ_𝐊(ℐ)·exp(∑_α∈ℐφ_α(x)),  ℐ∈𝒫_k[n]

then (<ref>) is of the form (<ref>),

    τ(x)     =    (𝐀·(x)·𝐊)
        =    ∑_ℐ∈𝒫_k[n]Λ_ℐ(x).

In this way, we move from degenerations multiplicities g
to more general products of minors g_ℐ. The dimension
k, which coincides with the rank of 𝐀 and 𝐊
when τ does not identically vanish, indicates the number of
line solitons at x_2≫0, while n-k is related to line solitons
at x_2≪0. In the statistical perspective, τ is the partition
function for a statistical model whose configurations correspond to
subsets ℐ∈𝒫_k[n] and have energies ∑_α∈ℐφ_α.
If κ_α∈ℤ and 𝐀∈ℤ_⋆^k× n
is a matrix with integer entries and maximal rank, then (<ref>)
is an integer, Δ_𝐀(ℐ) and Δ_𝐊(ℐ)
can be seen as degenerations for independent events and Δ_𝐀(ℐ)·Δ_𝐊(ℐ)
is the joint degeneration. The αth column of 𝐀
generalizes g_α in (<ref>),
so it can be regarded as a degeneration vector relative to the αth
energy level φ_α(x), α∈[n]. 

Regularity hypotheses for the case k=1 can be extended to τ-functions
at k≥3 too. For example, the entries of g in (<ref>)
and (<ref>) are assumed to be
positive since they are a measure for degenerations associated with
energy levels. At k>1, this property generalizes to a real matrix
𝐀∈ℝ_⋆^k× n where all the maximal
minors are non-negative. If one fixes the ordering κ_1<…<κ_n
for soliton parameters, then this request guarantees (indeed, it is
equivalent to, see e.g. <cit.>) the positivity of the
Cauchy-Binet expansion (<ref>),
hence the regularity of the solution of the original KP II equation.

In the next section, this kind of request will be relaxed: the matrix
𝐀 is only assumed to obey the full-rank condition. However,
some peculiarities of the total non-negative case will be discussed
in Section <ref>. 

 A choice of signs, or
signature, is a map 

    Σ: 𝔊⟶{±1}

where 

    𝔊:={ℐ∈𝒫_k[n]: Δ_𝐀(ℐ)≠0} .

Σ acts on (<ref>)
via g(ℐ)↦Σ(ℐ)· g(ℐ),
which returns a new function 

    ∑_ℐ∈𝒫_k[n]Σ(ℐ)· g_ℐ· e^∑_α∈ℐφ_α(x).


The set 𝔊 is a matroid on [n]: this means
that the exchange relation holds, i.e., 

    for  all  𝒜,ℬ∈𝔊, α∈𝒜∖ℬ, there  exists  β∈ℬ∖𝒜  such  that  𝒜_β^α∈𝔊.

In particular, we refer to a generic case as one where 𝔊=𝒫_k[n]. 



§  COMPLEXITY
REDUCTION THROUGH THE KP II CONSTRAINT


The determinantal partition function (<ref>)
at k=1 reduces to the statistical amoebas studied in <cit.>.
Cases at k>1 have more complications, due to the occurrence of
functional relations among the terms in the τ-function (<ref>).
Indeed, the minors of a (k× n)-matrix have to satisfy the
well-known Grassmann-Plücker relations (see, e.g., <cit.>,
3.1). In particular, the three-terms Plücker relations 

    Δ_𝐀(ℋ_αγ)·Δ_𝐀(ℋ_βδ)=Δ_𝐀(ℋ_αβ)·Δ_𝐀(ℋ_γδ)+Δ_𝐀(ℋ_αδ)·Δ_𝐀(ℋ_β,γ)

hold for all 1≤α<β<γ<δ≤ n and ℋ⊂[n]
with #ℋ=k-2 and {α,β,γ,δ}∩ℋ=∅. 

 For a generic matrix 𝐀∈ℝ_⋆^k× n,
k>1, not all the signatures (<ref>) for g∈ℝ^nk
in (<ref>) correspond to another
choice of matrix ∈ℝ^k× n
in (<ref>). In fact, consider 𝐀∈ℝ^k× n
such that there exist two non-vanishing minors Δ_𝐀(ℋ_αγ)
and Δ_𝐀(ℋ_βδ), e.g., taking
𝐀 parametrizing a point in the totally positive part of
the Grassmannian 𝒢ℛ_k,n(ℝ). Suppose that the
choice 

    g_ℐ↦{[  -g_ℐ, ℐ=ℋ_αγ;   g_ℐ,        ].,

which lies in the free statistical 1-amoeba relative to (<ref>),
corresponds to a certain matrix ∈ℝ^k× n.
The relation (<ref>) gives 

    Δ_𝐀(ℋ_αγ)·Δ_𝐀(ℋ_βδ)     =    Δ_𝐀(ℋ_αβ)·Δ_𝐀(ℋ_γδ)+Δ_𝐀(ℋ_αδ)·Δ_𝐀(ℋ_βγ)
        =    Δ_(ℋ_αβ)·Δ_(ℋ_γδ)+Δ_(ℋ_αδ)·Δ_(ℋ_βγ)
        =    Δ_(ℋ_αγ)·Δ_(ℋ_βδ) = -Δ_𝐀(ℋ_αγ)·Δ_𝐀(ℋ_βδ)

that means Δ_𝐀(ℋ_αγ)·Δ_𝐀(ℋ_βδ)=0,
a contradiction. It follows that, for points in the totally positive
Grassmannian, no signature of the type (<ref>)
preserves the form (<ref>). 

In general, it is not a trivial task to check if a given map of the
type (<ref>) follows from maximal minors of a
certain matrix. This issue is related to combinatorial structures
behind Grassmann-Plücker relations, namely chirotopes χ: [n]^k⟶{-1,0,+1}
(see, e.g., <cit.> for more details). In
particular, chirotopes coming from 𝐀∈ℝ^k× n,
in the sense that χ(ℐ)=sign(Δ_𝐀(ℐ))
for all ℐ∈[n]^k, are said to be realizable. Both
the check of the realizability of chirotopes and their enumeration
have non-trivial complexity <cit.>. However, in the present framework, the data on the initial function, in particular the “lengths” of the exponential and pre-exponential terms, are given:
starting from (<ref>), these absolute
values are known to be compatible with at least one determinantal
(Wronskian) form, and can be used to explore other signatures.

We exclude situations where (<ref>)
identically vanishes, so 𝐀 has maximal rank and there
is no null row 0_n^T. If there exists a null column,
then the τ-function does not depend on the corresponding soliton.
This leads to the reduction of a n- to a (n-1)-soliton solution.
So we can assume that there is no null column without loss of generality.

We now consider the conditions on choices of signs that map a τ-function
of the KP II equation (<ref>) to another τ-function, which will be called solitonic signatures. Let (<ref>) be any choice of signs for coefficients
g_ℐ in (<ref>),
which is equivalent to a choice of a partition of 𝔊
in two disjoint subsets 𝔊=∪,
∩=∅,
where 

    :=Σ^-1({+1}),  :=Σ^-1({-1}).

So one can write the resulting exponential sum as 

    ∑_ℐ∈𝒫_k[n]Σ(ℐ)·Δ_𝐀(ℐ)·Δ_𝐊(ℐ)· e^∑_α∈ℐφ_α(x)=τ(x)-2·τ_Σ(x)

where 

    τ_Σ(x):=∑_ℐ∈Δ_𝐀(ℐ)·Δ_𝐊(ℐ)· e^∑_α∈ℐφ_α(x).

Moreover, we will say 

    ℐ≅𝒥⇔Σ(ℐ)=Σ(𝒥),  ℐ,𝒥∈𝒫_k[n],

which defines a relation on 𝔊.

We do not assume a priori that a partition (<ref>)
returns a determinant (<ref>): this means that the
determinantal properties (i.e., Grassmann-Plücker relations) hold for
the family {Δ_𝐀(ℐ):. .ℐ∈𝔊},
but not necessarily for {Σ(ℐ)·Δ_𝐀(ℐ): ℐ∈𝔊}.

The bilinearity of the Hirota derivative and the KP operator (<ref>)
gives 

    D_KP(τ-2·τ_Σ,τ-2·τ_Σ)
        =    D_KP(τ,τ)+4·D_KP(τ_Σ,τ_Σ)-2·D_KP(τ,τ_Σ)-2·D_KP(τ_Σ,τ)
        =     4·D_KP(τ_Σ,τ_Σ)-2·D_KP(τ,τ_Σ)-2·D_KP(τ_Σ,τ)

since D_KP(τ,τ)=0. Therefore, one has 

    0=D_KP(τ,τ_Σ)-D_KP(τ_Σ,τ_Σ)=D_KP(τ-τ_Σ,τ_Σ).

We can say that τ-τ_Σ is “orthogonal” to τ_Σ
with respect to the D_KP bilinear operator. The bilinearity
also implies that (<ref>) is equivalent to

    0     =    D_KP(τ-τ_Σ,τ_Σ)
        =    ∑_𝒜∈∑_ℬ∈g(𝒜)g(ℬ)·D_KP[exp(∑_α∈𝒜∑_u=1^dκ_α^ux_u),exp(∑_β∈ℬ∑_u=1^dκ_β^ux_u)]
        =    ∑_𝒜∈∑_ℬ∈g(𝒜)g(ℬ)exp(2∑_β∈ℬ∑_u=1^dκ_β^ux_u)D_KP[1,exp(∑_u=1^d∑_α∈𝒜κ_α^ux_u-∑_β∈ℬκ_β^ux_u)].

From 

    D_KP[1,exp(∑_u=1^d∑_α∈𝒜κ_α^ux_u-∑_β∈ℬκ_β^ux_u)]
        =     (∂_x_1^4-4·∂_x_1∂_x_3+3·∂_x_2^2)exp(∑_u=1^d∑_α∈𝒜κ_α^ux_u-∑_β∈ℬκ_β^ux_u)
        =     C(𝒜,ℬ;)·exp(∑_u=1^d∑_α∈𝒜κ_α^ux_u-∑_β∈ℬκ_β^ux_u)

where 

    C(𝒜,ℬ;)     :=    [∑_α∈𝒜κ_α-∑_β∈ℬκ_β]^4+3·[∑_α∈𝒜κ_α^2-∑_β∈ℬκ_β^2]^2
           -4·[∑_α∈𝒜κ_α-∑_β∈ℬκ_β]·[∑_α∈𝒜κ_α^3-∑_β∈ℬκ_β^3].

The equation (<ref>) is equivalent to 

    ∑_𝒜∈∑_ℬ∈g(𝒜)g(ℬ)· C(𝒜,ℬ;)·exp(∑_u=1^d∑_α∈𝒜κ_α^ux_u+∑_β∈ℬκ_β^ux_u)=0.

Two exponentials in (<ref>) associated with
the pairs (𝒜,ℬ) and (𝒞,𝒟)
identically coincide if and only if 

    ∑_α∈𝒜κ_α^u+∑_β∈ℬκ_β^u=∑_γ∈𝒞κ_γ^u+∑_δ∈𝒟κ_δ^u,   u∈[d].

The only occurrences of (<ref>)
that are satisfied for a generic choice of the parameters 
involve the cases when 𝒜∪ℬ=𝒞∪𝒟
and 𝒜∩ℬ=𝒞∩𝒟. So,
if one assumes that there is no algebraic dependence that relates
these sums when 𝒜∪ℬ≠𝒞∪𝒟
or 𝒜∩ℬ≠𝒞∩𝒟, then
each exponential term in (<ref>)
is determined by the union 𝒜∪ℬ and the intersection
𝒜∩ℬ, since 

    ∑_u=1^d∑_α∈𝒜κ_α^ux_u+∑_β∈ℬκ_β^ux_u=∑_u=1^d∑_α∈𝒜∩ℬκ_α^u· x_u+∑_β∈𝒜∪ℬκ_β^u· x_u.

In particular, one has the following 

 Assume that the minors Δ_𝐀(ℐ)
with ℐ∈𝒫_k[n] are not vanishing, i.e., 𝔊=𝒫_k[n].
Then for each ℋ⊆ℒ⊆[n] with
#ℋ+4=#ℒ=k+2, the number of pairs (ℐ_+,ℐ_-)
associated with a term in (<ref>) with union
ℐ_+∪ℐ_-=ℒ and intersection
ℐ_+∩ℐ_-=ℋ is equal to (<ref>)
is 0 or 3. With the notation introduced in (<ref>)-(<ref>),
this can be stated as 

    ℐ_γδ^αβ≅ℐ⇔ℐ_γ^α≅ℐ_δ^β⇔ℐ_δ^α≅ℐ_γ^β.



Suppose that there exist subsets ℐ_+∈,
ℐ_-∈ such that ℋ:=ℐ_+∩ℐ_-=:{γ_1,…,γ_k-2},
γ_1<…<γ_k-2, and ℒ:=ℐ_+∪ℐ_-,
so the number of terms in (<ref>) associated
with ℋ and ℒ is not 0. In particular,
#(ℐ_+Δℐ_-)=4. Then, there exist α_+,β_+∈ℐ_+
and α_-,β_-∈ℐ_- such that ℐ_±=ℋ∪{α_±,β_±}.
There are two additional pairs of subsets different from {ℐ_+,ℐ_-}
with the same union and intersection. Thus, in order to have a vanishing
coefficient for the associated term in (<ref>),
at least one of these two pairs has to belong to ×.
Let us suppose that only one of these two possibilities is in ×,
call it (ℒ_+,ℒ_-) with ℒ_±=ℋ∪{α_±,β_∓}.
In such a case, the term C(ℐ_+,ℐ_-;)
in (<ref>) is 

    C(ℐ_+,ℐ_-;)=12·(κ_α_+-κ_α_-)(κ_β_+-κ_α_-)(κ_α_+-κ_β_-)(κ_β_+-κ_β_-)

that is not vanishing since the soliton parameters are pairwise distinct
by assumption. Using the parity P((β_-,α_-,β_+,α_+))
of the permutation (β_-,α_-,β_+,α_+),
we introduce 

    σ(α_+,β_+|α_-,β_-)     :=    sign[(α_+-α_-)(β_+-α_-)(α_+-β_-)(β_+-β_-)]
        =    P((β_-,α_-,β_+,α_+))·sign[(α_+-β_+)(α_--β_-)]

which gives 

    Δ_𝐊(ℐ_+)·Δ_𝐊(ℐ_-)· C(ℐ_+,ℐ_-;)
        =     12·∏_α_i<α_j^ℐ_+(κ_α_j-κ_α_i)·∏_β_i<β_j^ℐ_-(κ_β_j-κ_β_i)·(κ_α_+-κ_α_-)(κ_β_+-κ_α_-)(κ_α_+-κ_β_-)(κ_β_+-κ_β_-)
        =     12·∏_γ_i<γ_j^ℐ_+∩ℐ_-(κ_γ_j-κ_γ_i)·∏_δ_i<δ_j^ℐ_+∪ℐ_-(κ_δ_j-κ_δ_i)·σ(α_+,β_+|α_-,β_-)
        =     12·VdM(;ℐ_+∩ℐ_-)·VdM(;ℐ_+∪ℐ_-)·σ(α_+,β_+|α_-,β_-).

The signs σ(α_+,β_-|α_-,β_+) and
σ(α_+,α_-|β_+,β_-) can be found
in the same way, taking into account that 

    P((β_-,α_-,β_+,α_+))=-P((β_-,β_+,α_-,α_+))=-P((β_+,α_-,β_-,α_+)).

Furthermore, one can consider 

    S(α,ℋ):=max{i∈[k]: γ_i-1<α},  α∈[n]\ℋ

where γ_0:=0. In this way, it is easy to check that 

    Δ_𝐀(ℋ_αβ)=Δ_𝐀(α,β;ℋ)·(-1)^1+S(α,ℋ)+S(β,ℋ)·sign[α-β].

where Δ_𝐀(α,β;ℋ) is the product
of Δ_𝐀(ℋ_αβ) and the parity
of the permutation (α,β,γ_1,…,γ_k-2).
Thus, one gets 

    g(ℐ_+)· g(ℐ_-)· C_ℐ_+ℐ_-()
        =     12·VdM(;ℐ_+∩ℐ_-)·VdM(;ℐ_+∪ℐ_-)·P((β_-,α_-,β_+,α_+))
       ·   (Δ_𝐀(ℋ_α_+β_+)·Δ_𝐀(ℋ_α_-β_-)·sign[(α_+-β_+)(α_--β_-)])
        =    { 12·VdM(;ℐ_+∩ℐ_-)·VdM(;ℐ_+∪ℐ_-)·(-1)^S(α_+,ℋ)+S(β_+,ℋ)+S(α_-,ℋ)+S(β_-,ℋ)}
       ·   {P((β_-,α_-,β_+,α_+))·Δ_𝐀(α_+,β_+|ℋ)·Δ_𝐀(α_-,β_-|ℋ)} .

The first term in braces in (<ref>) is the same for
all the pairs (𝒢_+,𝒢_-) with 𝒢_+∩𝒢_-=ℋ
and 𝒢_+∪𝒢_-=ℒ. For the second
term in braces, the antisymmetry (<ref>)
implies that 

    g(ℒ_+)· g(ℒ_-)· C_ℒ_+ℒ_-()
        =    [12·VdM(;ℐ_+∩ℐ_-)·VdM(;ℐ_+∪ℐ_-)·(-1)^S(α_+,ℋ)+S(β_+,ℋ)+S(α_-,ℋ)+S(β_-,ℋ)]
       ·   [-P((β_-,α_-,β_+,α_+))·Δ_𝐀(α_+,β_-|ℋ)·Δ_𝐀(α_-,β_+|ℋ)].

The three-terms Plücker relations (<ref>),
which are valid for minors Δ_𝐀(ℐ), can
be stated for any four pairwise distinct elements δ_a∈[n]\ℋ,
a∈[4], as 

    Δ_𝐀(δ_1,δ_2|ℋ)·Δ_𝐀(δ_3,δ_4|ℋ)=Δ_𝐀(δ_1,δ_3|ℋ)·Δ_𝐀(δ_2,δ_4|ℋ)-Δ_𝐀(δ_1,δ_4|ℋ)·Δ_𝐀(δ_2,δ_3|ℋ).

In particular, if one looks at the sum of (<ref>)
and (<ref>) and applies (<ref>)
with (δ_1,δ_2,δ_3,δ_4)≡(α_+,α_-,β_+,β_-),
the result is 

    g(ℐ_+)· g(ℐ_-)· C_ℐ_+ℐ_-()+g(ℒ_+)· g(ℒ_-)· C_ℒ_+ℒ_-()
        =     12·VdM(;ℐ_+∩ℐ_-)·VdM(;ℐ_+∪ℐ_-)
       ·    (-1)^S(α_+,ℋ)+S(β_+,ℋ)+S(α_-,ℋ)+S(β_-,ℋ)·P((β_-,α_-,β_+,α_+))
       ·   (Δ_𝐀(α_+,β_+|ℋ)·Δ_𝐀(α_-,β_-|ℋ)-Δ_𝐀(α_+,β_-|ℋ)·Δ_𝐀(α_-,β_+|ℋ))
        =     12·VdM(;ℐ_+∩ℐ_-)·VdM(;ℐ_+∪ℐ_-)·(-1)^S(α_+,ℋ)+S(β_+,ℋ)+S(α_-,ℋ)+S(β_-,ℋ)
       ·   P((β_-,α_-,β_+,α_+))·Δ_𝐀(α_+,α_-|ℋ)·Δ_𝐀(β_+,β_-|ℋ)≠0

since we have assumed that all the minors are not vanishing. Hence,
the coefficient in (<ref>) associated with
ℋ and ℒ is not vanishing and the associated
τ-function is not a solution of the KP equation (<ref>).

On the contrary, if all the three terms are involved in (<ref>),
then their sum is 

    12·VdM(;ℐ_+∩ℐ_-)·VdM(;ℐ_+∪ℐ_-)
       ·    (-1)^S(α_+,ℋ)+S(β_+,ℋ)+S(α_-,ℋ)+S(β_-,ℋ)·P((β_-,α_-,β_+,α_+))
       ·   (Δ_𝐀(α_+,β_+|ℋ)·Δ_𝐀(α_-,β_-|ℋ)-Δ_𝐀(α_+,β_-|ℋ)·Δ_𝐀(α_-,β_+|ℋ).
          .-Δ_𝐀(α_+,α_-|ℋ)·Δ_𝐀(β_+,β_-|ℋ))=0

since the second term in square brackets vanishes due to the three-terms
Plücker relations (<ref>). 



Let ℐ∈𝒫_k[n], α_1,α_2∈ℐ
and δ_1,δ_2∈[n]∖ℐ such that ℐ,ℐ_δ_1δ_2^α_1α_2∈𝔊.
Then at least one of the two products Δ_𝐀(ℐ_δ_T^α_1)·Δ_𝐀(ℐ_δ_U^α_2),
{T,U}={1,2}, is not vanishing. Further, if Δ_𝐀(ℐ_δ_T^α_1)·Δ_𝐀(ℐ_δ_U^α_2)≠0,
then Σ(ℐ_δ_T^α_1)·Σ(ℐ_δ_U^α_2)=Σ(ℐ)·Σ(ℐ_δ_1δ_2^α_1α_2). 


First suppose that one of the product vanishes, e.g., Δ_𝐀(ℐ_δ_2^α_1)·Δ_𝐀(ℐ_δ_1^α_2)=0
without loss of generality. Then the three-terms Plücker relations
(<ref>) imply that |Δ_𝐀(ℐ_δ_2^α_1)·Δ_𝐀(ℐ_δ_1^α_2)|=|Δ_𝐀(ℐ)·Δ_𝐀(ℐ_δ_1δ_2^α_1α_2)|≠0.
Further, Σ(ℐ_δ_T^α_1)·Σ(ℐ_δ_U^α_2)≠Σ(ℐ)·Σ(ℐ_δ_1δ_2^α_1α_2)
implies that there is exactly one non-vanishing term in (<ref>)
associated with a pair in ×
with intersection ℐ^α_1α_2 and union
ℐ_δ_1δ_2, hence (<ref>)
do not vanish. Thus Σ(ℐ_δ_T^α_1)·Σ(ℐ_δ_U^α_2)=Σ(ℐ)·Σ(ℐ_δ_1δ_2^α_1α_2).
If instead both Δ_𝐀(ℐ_δ_1^α_1)·Δ_𝐀(ℐ_δ_2^α_2)
and Δ_𝐀(ℐ_δ_2^α_1)·Δ_𝐀(ℐ_δ_1^α_2)
are not vanishing, then (<ref>) holds
because of Lemma <ref> (see in particular
(<ref>) and (<ref>)).  


 Let ℋ,𝒦∈𝔊
with r:=#(ℋ\𝒦). Then there exists
a finite sequence ℒ_0:=ℋ, ℒ_1, …,ℒ_r:=𝒦
of elements of 𝔊 such that #(ℒ_u-1Δℒ_u)=2,
u∈[r]. 


Let ℋ\𝒦=:{γ_1,…,γ_r}
and ℒ_0:=ℋ, so the exchange property (<ref>)
implies that there exists Ψ(γ_1)∈𝒦∖ℋ
such that ℒ_1:=ℋ∖{γ_1}∪{Ψ(γ_1)}∈𝔊.
Note that #(ℒ_1Δ𝒦)=r-1. One can iterate
the process, starting from ℒ_u∈𝔊 and finding
Ψ(γ_u+1)∈𝒦∖ℒ_u in order
to define 

    ℒ_u+1:=ℋ\{γ_1,…,γ_u+1}∪{Ψ(γ_1),…,Ψ(γ_u+1)},   u∈[r]

such that ℒ_u+1∈𝔊 too. At each step, the
element Ψ(γ_u+1) is different from Ψ(γ_t)
for all t≤ u because Ψ(γ_t)∈ℒ_u,
while Ψ(γ_u+1)∈𝒦∖ℒ_u.
So the map Ψ: ℋ\𝒦⟶𝒦\ℋ
is injective and, hence, a bijection, since #ℋ=k=#𝒦
implies #(ℋ\𝒦)=#(𝒦\ℋ).
All the subsets ℒ_0,ℒ_1,…,ℒ_r
are elements of 𝔊 and #(ℒ_u-1Δℒ_u)=2
for all u∈[r].  

Now we consider the following relations 

    α≈_𝒜β⇔𝒜≅𝒜_α^β,  α∈[n]∖ℐ, β∈ℐ.

If there is λ∈𝒜 such that ℐ_λ^α,ℐ_λ^β∈𝔊
where ℐ=𝒜_α^λ, then one can
express α≈_ℐβ as ℐ_λ^α≅ℐ_λ^β.
This does not depend on the choice of λ∈[n]\ℐ,
as can be easily shown: 

 Let ℐ∈𝒫_k[n]
and α,β∈ℐ. Then ℐ_γ^α≅ℐ_γ^β
holds for a certain γ∈[n]\ℐ if and only
if ℐ_δ^α≅ℐ_δ^β
holds for all δ∈[n]\ℐ such that ℐ_δ^α,ℐ_δ^β∈𝔊. 


One implication is trivial. So assume that there exist ℐ∈𝒫_k[n],
α,β∈ℐ and γ,δ∈[n]\ℐ
such that ℐ_γ^α≅ℐ_γ^β
and ℐ_δ^α≅-ℐ_δ^β
(clearly γ≠δ). The case 0∈{ℐ_δ^α,ℐ_δ^β}
is ruled out by the assumption ℐ_δ^α,ℐ_δ^β∈𝔊.
If ℐ_γ^α≅ℐ_δ^β,
then ℐ_δ^α≅ℐ_γ^β
by Proposition <ref>.
This means that ℐ_δ^β≅ℐ_γ^α≅ℐ_γ^β≅ℐ_δ^α≅-ℐ_δ^β,
i.e., a contradiction. Now suppose that ℐ_γ^α≅-ℐ_δ^β,
which means ℐ_δ^α≅-ℐ_γ^β
by Proposition <ref>.
Thus ℐ_δ^β≅-ℐ_γ^α≅-ℐ_γ^β≅ℐ_δ^α≅-ℐ_δ^β,
i.e., a contradiction.  


In fact, the relations (<ref>) are
compatible for different choices of 𝒜 too. 

 There are no subsets
𝒜,ℬ∈𝒫_k[n], α∈[n]∖(𝒜∪ℬ),
and β∈𝒜∩ℬ such that 𝒜,𝒜_α^β,ℬ,ℬ_α^β∈𝔊
and Σ(𝒜)·Σ((𝒜)_α^β)=-Σ(ℬ)·Σ((ℬ)_α^β). 


Introduce 𝒜_1:=𝒜, ℬ_1:=ℬ,
𝒜_2:=(𝒜)_α^β, ℬ_2:=(ℬ)_α^β.
Note that 𝒜_1∖ℬ_1=𝒜_2∖ℬ_2,
ℬ_1∖𝒜_1=ℬ_2∖𝒜_2
and α,β∉𝒜_1Δℬ_1.

We shall prove the statement by induction on the distance between
𝒜_1 and ℬ_1, that is r:=1/2·#(𝒜_1Δℬ_1).
First consider the case r=1 and set ℬ_1∖𝒜_1={Ψ(γ_1)}.
In particular, the elements α,β,γ_1,Ψ(γ_1)
are all distinct and the set ℋ:=𝒜_1∩𝒜_2∩ℬ_1∩ℬ_2∈𝒫_k-2[n]
satisfies 𝒜_1=ℋ_βγ_1, 𝒜_2=ℋ_αγ_1,
ℬ_1=ℋ_βΨ(γ_1) and ℬ_2=ℋ_αΨ(γ_1).
By hypothesis, 𝒜_1,𝒜_2,ℬ_1,ℬ_2∈𝔊,
so Proposition <ref>
gives Σ(𝒜_1)·Σ(ℬ_2)=Σ(𝒜_2)·Σ(ℬ_1).
Multiplying both sides of this equation by Σ(𝒜_2)·Σ(ℬ_2)
and applying Σ(𝒜_2)^2=Σ(ℬ_2)^2=1
we find 

    Σ(𝒜_1)·Σ(𝒜_2)=Σ(ℬ_1)·Σ(ℬ_2).

Now assume that the statement holds whenever 1/2·#(𝒜Δℬ)<r
and take 𝒜_1,𝒜_2, ℬ_1,ℬ_2∈𝔊
with 1/2·#(𝒜_1Δℬ_1)=r.
From Lemma <ref> there exist a labelling
𝒜_1∖ℬ_1={γ_1,…,γ_r}
and two bijections Ψ_T: 𝒜_T∖ℬ_T⟶ℬ_T∖𝒜_T,
T∈{1,2}, such that 

    ℒ_T^(u):=(𝒜_T)_Ψ_T(γ_1),…,Ψ_T(γ_u)^γ_1…γ_u∈𝔊,   u∈[r], T∈{1,2}.

Let us focus on the four sets 𝒜_1, 𝒜_2,
ℒ_1^(r-1), ℒ_2^(r-1): 𝒜_1,𝒜_2∈𝔊
by hypothesis, and ℒ_1^(r-1),ℒ_2^(r-1)∈𝔊
by construction. Further, one has β∈(𝒜_1∩ℒ_1^(r-1))∖(𝒜_2∩ℒ_2^(r-1))
and α∈(𝒜_2∩ℒ_2^(r-1))∖(𝒜_1∩ℒ_1^(r-1)).
Since the distance between ℒ_T^(u) and 𝒜_T
increases by 1 at each step, one gets 1/2·#(𝒜_TΔℒ_T^(r-1))=r-1.
So the inductive hypothesis applies and 

    Σ(𝒜_1)·Σ(ℒ_1^(r-1))=Σ(𝒜_2)·Σ(ℒ_2^(r-1)).

Likewise, ℒ_T^(r)=ℬ_T , β∈(ℒ_1^(r-1)∩ℬ_1)∖(ℒ_2^(r-1)∩ℬ_2)
and α∈(ℒ_2^(r-1)∩ℬ_2)∖(ℒ_1^(r-1)∩ℬ_1),
so one can repeat the same argument as in (<ref>)
and get 

    Σ(ℒ_1^(r-1))·Σ(ℬ_1)=Σ(ℒ_2^(r-1))·Σ(ℬ_2).

Multiplying (<ref>) and (<ref>)
side by side, and applying Σ(ℒ_1^(r-1))^2=+1,
one finds 

    Σ(𝒜_1)·Σ(ℬ_1)=Σ(𝒜_2)·Σ(ℬ_2)

that means α≈_𝒜β if and only if α≈_ℬβ. 


Hence, for each α,β∈[n], the relation (<ref>)
depends only on α and β, and not on ℐ
such that ℐ,ℐ_α^β∈𝔊.
If such a ℐ exists, we can introduce 

    χ(α,β)=Σ(ℐ)·Σ(ℐ_α^β)

without ambiguity. Moreover, χ is symmetric in its arguments,
since Σ(ℐ)·Σ(ℐ_α^β)=Σ(ℐ_α^β)·Σ((ℐ_β^α)_α^β)
and one can substitute ℐ↦ℐ_β^α
in (<ref>).

We will see that the signs (<ref>)
characterize Σ, so we focus on these pairs. 

 For each ℐ:={β_1,…,β_k}∈𝔊
take the k×(n-k) matrix 𝐗(ℐ) defined
as (𝐗(ℐ))_iα:=Σ(ℐ)·Σ(ℐ_α^β_i)
if ℐ_α^β_i∈𝔊 and 0 otherwise
(α∈[n]∖ℐ). We refer to a path
Φ_ℐ on 𝐗(ℐ) as a sequence
of non-vanishing entries obtained moving alternately along rows and
columns of 𝐗(ℐ), i.e., of the form 

    (i_1,α_1)→(i_2,α_1)→(i_2,α_2)→(i_3,α_2)→…

with i_T≠ i_T+1, α_T≠α_T+1 and ℐ^i_T_α_T,ℐ^i_T+1_α_T∈𝔊
for all T.
Two indices α,β∈[n] are said to be connected
by a path Φ_ℐ if α is a component of the
first element of Φ_ℐ but not of the second, and
β is a component of the last element of Φ_ℐ
but not of the second-to-last. 

For instance, if α∈ℐ and β∈[n]∖ℐ,
then a path from α to β starts with (α,γ)
and ends with (δ,β), for some γ∈[n]∖ℐ,δ∈ℐ;
if instead both α and β are elements of ℐ,
a path starts with (α,γ) and ends with (β,δ),
γ,δ∈[n]∖ℐ. These paths induce a relation
→_ℐ on [n]. 

 For all α,β∈[n],
we say α→_ℐβ if α=β
or there is a path in 𝐗(ℐ) that connects α
and β. 

This is an equivalence relation: it is reflexive by definition; if
α→_ℐβ, then the reverse path gives
β→_ℐα, so →_ℐ
is symmetric; it is also transitive, as follows from the path α→_ℐγ
obtained from the concatenation of α→_ℐβ
and β→_ℐγ and the simplification
of consecutive reverse subpaths. Moreover, the following result holds: 

 The product
of signs of edges χ along any closed path of 𝐗(ℐ),
ℐ∈𝔊, is +1. 


Let ℐ:={γ_1,…,γ_k} and denote γ_i_T
by i_T in the rest of the proof for the sake of clearness. The
same number of moves along rows and along columns is required to close
a path, so it has an even length 2r. Thus, the product of signs
along the closed path is 

    Φ_ℐ(i_1,…,i_r|α_1,…,α_r):=∏_T=1^rΣ(ℐ_α_T^i_T)·Σ(ℐ_α_T^i_T+1)

where indices T are taken modulo r, e.g., i_r+1=i_1. From
this one can also see that the statement is invariant under the replacement
of 𝐗(ℐ) by 𝐗(𝒥)∈𝔊,
as long as all the involved subsets 𝒥_α_T^i_T
and 𝒥_α_T^i_T+1 are in 𝔊:
indeed, both sides of (<ref>) can be multiplied
by (Σ(ℐ))^2r·(Σ(𝒥))^2r=+1,
so the factors are expressed as Σ(ℐ_α_T^i_T)·Σ(ℐ)=Σ(𝒥_α_T^i_T)·Σ(𝒥)
(respectively Σ(ℐ_α_T^i_T+1)·Σ(ℐ)=Σ(𝒥_α_T^i_T+1)·Σ(𝒥))
by Proposition <ref>.

We prove the statement by induction on r. The base cases are r=1,
which is trivial by symmetry of χ(α,γ_i) with respect
to the interchange of its arguments, and r=2. In the latter situation,
there exist i≠ m and α≠β such that ℐ_α^γ_i,ℐ_α^γ_m,ℐ_β^γ_m,ℐ_β^γ_i∈𝔊.
By Proposition <ref>
the associated product of signs is +1. Now assume that the thesis
holds for s≤ r-1 and take any path of length 2r. First suppose
that there exist T∈[r], S∈[r-2], such that ℐ_α_T+S^i_T∈𝔊.
Then one can write 

    Φ_ℐ(i_1,…,i_r|α_1,…,α_r)
        =    [(∏_W=0^S-1Σ(ℐ_α_T+W^i_T+W)·Σ(ℐ_α_T+W^i_T+W+1))·Σ(ℐ_α_T+S^i_T+S)·Σ(ℐ_α_T+S^i_T)]
       ·   [Σ(ℐ_α_T+S^i_T)·(∏_W=S^r-2Σ(ℐ_α_T+W^i_T+W+1)·Σ(ℐ_α_T+W+1^i_T+W+1))·Σ(ℐ_α_T+r-1^i_T)]
        =    Φ_ℐ(i_T,i_T+1,…,i_T+S|α_T,α_T+1,…,α_T+S)
       ·   Φ_ℐ(i_T,i_T+S+1,i_T+S+2,…,i_T+r-1|α_T+S,α_T+S+1,…,α_T+r-1).

The lengths of these two closed path are 2S+2 and 2(r-S) respectively,
which lie in {4,…,2r-2}; so the inductive hypothesis applies
to both these paths and this gives the result.

On the other hand, if such T,S do not exist, then ℐ_α_T^i_T+S∈𝒫_k[n]∖𝔊
for all T∈[r] and S∉{1,r}: by Proposition <ref>,
the condition ℐ_α_T^i_T,ℐ_α_T+S^i_T+S∈𝔊,
along with ℐ_α_T+S^i_T∈𝒫_k[n]∖𝔊
at S∉{r-1,r} and ℐ_α_T^i_T+S∈𝒫_k[n]∖𝔊
at S∉{1,r}, implies that ℐ_α_Tα_T+S^i_Ti_T+S∈𝔊
for all S≠ r. Likewise, from ℐ_α_T^i_T+1,ℐ_α_T+S^i_T+S+1∈𝔊,
ℐ_α_T+S^i_T+1∈𝒫_k[n]∖𝔊
(S∉{1,r}) and ℐ_α_T^i_T+S+1∈𝒫_k[n]∖𝔊
(S∉{r-1,r}), one gets ℐ_α_Tα_T+S^i_T+1i_T+S+1∈𝔊
whenever S≠ r. Further, ℐ_α_2^i_2,ℐ_α_T^i_T+1∈𝔊
and ℐ_α_2^i_T+1∈𝒫_k[n]∖𝔊
lead to ℐ_α_2α_T^i_2i_T+1∈𝔊
for all T∉{1,2}.

So we fix 𝒥:=ℐ_α_2^i_2∈𝔊:
by previous observations, one has 𝒥_α_T^i_T,𝒥_α_T^i_T+1∈𝔊
for all T∉{1,2}, 𝒥_α_1^i_1∈𝔊
and 𝒥_α_1^i_3=ℐ_α_1α_2^i_2i_3∈𝔊.
One can apply Proposition <ref>
to 𝒥_α_1^i_1 and (𝒥_α_1^i_1)_i_1^i_3=𝒥_α_1^i_3,
which are in 𝔊, and write 

    Σ(𝒥_α_1^i_1)·Σ(𝒥_α_1^i_3)     =    Σ(ℐ_α_1α_2^i_1i_2)·Σ(ℐ_α_1α_2^i_2i_3)·Σ(ℐ)^2
        =    Σ(ℐ_α_1^i_1)·Σ(ℐ_α_2^i_2)·Σ(ℐ_α_1^i_2)·Σ(ℐ_α_2^i_3)

Thus consider the path Φ_𝒥(i_1,i_3,i_4,…,i_r|α_1,α_3,α_4,…,α_r)
of length 2(r-1) on 𝐗(𝒥), which also satisfies
Σ(𝒥_α_T^i_T)·Σ(𝒥_α_T^i_T+1)=Σ(ℐ_α_T^i_T)·Σ(ℐ_α_T^i_T+1)
for all T∉{1,2} (all the subsets are in 𝔊
and Proposition <ref> holds)
and Σ(𝒥_α_1^i_1)·Σ(𝒥_α_1^i_3)=Σ(ℐ_α_1^i_1)·Σ(ℐ_α_1^i_2)·Σ(ℐ_α_2^i_2)·Σ(ℐ_α_2^i_3)
by (<ref>). Since the involved subsets
are in 𝔊, the inductive hypothesis applies to the path
in 𝐗(𝒥) and that gives 

    Φ_ℐ(i_1,…,i_r|α_1,…,α_r)=Φ_𝒥(i_1,i_3,…,i_r|α_1,α_3,…,α_r)=+1.
 


A special role is assumed by the pivot set 𝒱, since the
hypothesis of no-null columns implies that each α∈[n] is
associated with an element ν_i∈𝒱 (possibly α=ν_i)
such that 𝒱_α^ν_i∈𝔊. So we
can finally state the main result: 


A choice of signs Σ: 𝔊⟶{±1}
returns a solution of the KP II equation (<ref>)
if and only if Σ is induced by a choice of signs for rows
and columns of 𝐀 (up to the action of GL_k(ℝ)). 


One implication is trivial, since a choice of signs for rows and columns
of 𝐀 induces a (singular) soliton solution by construction.
So consider any choice of signs Σ that returns a solution
of the KP II equation and choose an arbitrary element in 𝔊,
e.g., 𝒱. For each ℐ∈𝔊 and α∈ℐ∖𝒱
, given that Δ_𝐀(ℐ)·Δ_𝐀(𝒱)≠0,
the Plücker relations imply that there exists at least one non-vanishing
term of the type Δ_𝐀(ℐ_ν_i^α)·Δ_𝐀(𝒱_α^ν_i).
Thus one can always find ν(α)∈𝒱∖ℐ
such that both ℐ_ν(α)^α and 𝒱_α^ν(α)
are elements of 𝔊. So let ℐ∖𝒱={α_1,…,α_r}:
as in Lemma <ref>, we start with ℒ_0:=ℐ
and, from ℒ_u-1∈𝔊 and α_u∈ℒ_u-1∖𝒱,
u∈[r], we find 

    ν(α_u)∈𝒱∖ℒ_u-1,  ℒ_u:=ℒ_u-1\{α_u}∪{ν(α_u)}

such that ℒ_u,𝒱_α_u^ν(α_u)∈𝔊.
Also in this case, from ν(α_s)∈ℒ_u-1 for
all s<u and ν(α_u)∈𝒱∖ℒ_u-1,
it follows that all ν(α_u) are pairwise distinct. So we
get 

    Σ(ℐ)·Σ(𝒱)     =    ∏_u=1^rΣ(ℒ_u-1)·Σ(ℒ_u)=∏_u=1^rΣ(ℒ_u-1)·Σ((ℒ_u-1)_ν(α_u)^α_u)
        =    ∏_u=1^rΣ(𝒱_α_u^ν(α_u))·Σ((𝒱_α_u^ν(α_u))_ν(α_u)^α_u)  (from  Proposition  <ref>)
        =    ∏_u=1^rΣ(𝒱_α_u^ν(α_u))·Σ(𝒱)=∏_u=1^rχ(α_i,ν(α_i)).


Now consider the equivalence →_𝒱. Each class
𝒞_p contains at least one element ν_i_p∈𝒱,
since we have assumed that there are no null columns: fix a sign χ(ν_i_p)∈{±1}
for each of them. For any α∈𝒞_p, take
a path Φ(i_p→α) on 𝐗(𝒱)
connecting ν_i_p and α, and set 

    χ(α):=χ(i_p)·(∏_(m,δ)∈Φ(i_p→α)χ(δ,ν_m)).

This definition is well-posed since it does not depend on the choice
of the path by Proposition <ref>.
If 𝒱_α^ν_i∈𝔊, then ν_i∈𝒱
belongs to the same class of α, because the path with only
one element (ν_i,α) connects them. The concatenation of
{(ν_i,α)}, the reverse of Φ(i_p→α)
and Φ(i_p→ν_i) makes a closed path, whose product
of signs is equal to +1 by Proposition <ref>.
So 

    χ(α,ν_i)     =    ∏_(m,δ)∈Φ(i_p→α)χ(δ,ν_m)·∏_(l,γ)∈Φ(i_p→ν_i)χ(γ,ν_l)
        =    (χ(i_p)·∏_(m,δ)∈Φ(i_p→α)χ(δ,ν_m))·(χ(i_p)·∏_(l,γ)∈Φ(i_p→ν_i)χ(γ,ν_l))
        =    χ(α)·χ(ν_i).

Finally, since 𝒱∖ℐ={ν(α_1),…,ν(α_r)},
we can express Σ(ℐ) as 

    Σ(ℐ)     =    Σ(𝒱)·∏_u=1^rχ(α_u,ν(α_u))  (from  (<ref>))
        =    Σ(𝒱)·∏_u=1^rχ(α_u)·χ(ν(α_u))  (from  (<ref>))
        =    Σ(𝒱)·(∏_i=1^kχ(ν_i))·(∏_i=1^kχ(ν_i))·(∏_u=1^rχ(ν(α_u)))·(∏_u=1^rχ(α_u))
        =     R·(∏_ν_i∈ℐ∩𝒱χ(ν_i))·(∏_u=1^rχ(α_u))=R·∏_α∈ℐχ(α).

where 

    R:=Σ(𝒱)·(∏_i=1^kχ(ν_i)).

Hence Σ is induced by a choice of sign χ(α) for
columns α∈[n] and R for an arbitrary row of 𝐀.


The previous results can be summarised in the following theorem,
which relates the requirements coming from the determinantal form
(<ref>), the KP II equation
and the whole KP hierarchy. 

 For
a generic choice of , a signature (<ref>)
returns a solution τ(x)-2·τ_Σ(x)
of the KP II equation (<ref>) if and only if it returns
a solution τ(x_1,x_2,x_3,…)-2·τ_Σ(x_1,x_2,x_3,…)
for the whole KP hierarchy, if and only if Σ preserves the determinant form (<ref>). 


A choice of signs (<ref>) returns a solution
of the KP II equation (<ref>) if and only if it is
induced by a choice of signs for rows and columns of 𝐀
by Theorem <ref>.
In this case, the associated function τ(x_1,x_2,x_3,…)-2·τ_Σ(x_1,x_2,x_3,…) is a solution of the whole KP hierarchy. 

If we focus on the determinantal structure of the τ-function, all the choices of signs for rows and columns clearly preserve the form (<ref>). On the other hand, if a signature Σ preserve this structure, then the KP II equation is satisfied (together with all the other members of the hierarchy). So, from Theorem <ref>, this solution can be expressed in terms of the initial function via a choice of signs for rows and columns of 𝐀.  
 
A more detailed analysis on the signatures preserving the determinantal constraints, which also includes the preservation of a specific subset of soliton parameters, is given in Appendix <ref>. 

A special situation is when Σ is defined over the whole set
𝒫_k[n], i.e., 𝔊=𝒫_k[n]. In
such a case, one can easily verify that Remark <ref>
implies that ≈_ℐ is an equivalence. If some vanishing
minors occur, then the transitivity of the relation (<ref>)
is not guaranteed. So one could look for a transitive extension of
all the relations ≈_ℐ at varying ℐ.
An extension of Σ, that
is a map Σ̃: 𝒫_k[n]⟶{±1}
that satisfies Σ̃(ℐ)=Σ(ℐ)
for all ℐ∈𝔊, can be obtained using (<ref>)
to label minors in 𝒫_k[n]∖𝔊 with
a sign compatible with Proposition <ref>.
So the following holds: 

 For a solitonic choice
of sign Σ, there exists an extension Σ̃ of
Σ to the whole set 𝒫_k[n] such that Σ̃
is a solitonic choice of signs too. 

It could be interesting to extend this approach to more general expressions
for the τ-function and to other hierarchies, in order to check
the complexity reduction coming from the initial data and the specific
requirements.



§  NUMBER OF DISTINCT
CONFIGURATIONS


Having identified a family of signatures that preserve specific requirements,
it is worth exploring some of its combinatorial aspects in order to
clarify the effects of the initial data, e.g., the coefficient matrix
𝐀, on the set of allowed configurations.

The overall sign given by row signature is obtained with the choice
R∈{±1} for an arbitrary row of 𝐀 (see (<ref>)).
Sign flips for columns can be expressed as an action of {±1}^[n]
on ℝ^k× n by right multiplication, i.e., _α(𝐀):=𝐀·_α
and _ℐ(𝐀):=𝐀·_ℐ
where 

    _α    :=    (
    [ _α-1    0    0;    0   -1    0;    0    0 _n-α ]),  α∈[n],
    _𝒮    :=    ∏_α∈𝒮_α,  𝒮∈𝒫[n].

Accordingly, (<ref>) becomes

    τ_𝒮(x):=(_𝒮(𝐀)·(x)·𝐊).


If k=1, then the minors of A∈ℝ^1× n
are the entries of a row vector. Then, for every non-zero matrix B∈ℝ^1× n
such that |A_α|=|B_α| for all α∈[N], one
can recover A from (<ref>)
choosing 𝒮={α∈[n]: A_1,α· B_1,α<0}.
If 𝔊=𝒫_k[n], such a choice is unique. The
use of (<ref>) in the case k>1 gives
a suitable generalization. 

 Let 𝐀∈ℝ^k× n
and 𝒞_1,…,𝒞_P be the equivalence classes
associated with the relation →_𝒱 in Definition
<ref>. For all ℋ_1,ℋ_2⊆[n],
the equalities 

    sign[Δ(_ℋ_1(𝐀);ℐ)]=sign[Δ(_ℋ_2(𝐀);ℐ)],  ℐ∈𝔊

imply that there exists 𝔭⊆[P] such that 

    ℋ_1Δℋ_2=⋃_q∈𝔭𝒞_q.



First note that the condition (<ref>)
holds for 𝐀 if and only if it holds for _ℋ_2(𝐀):
indeed, for all 𝒮⊆[n], 𝐀 and _𝒮(𝐀)
have the same set of non-vanishing maximal minors and, in particular,
the action of _ℋ_2 does
not affect the relation →_𝒱 relative to the
pivot set 𝒱. In the substitution 𝐀↦_ℋ_2(𝐀),
the additional signs coming from _α,
α∈ℋ_2, appear on both sides of (<ref>),
then 

    _ℋ_1(𝐀)    ↦   _ℋ_1(_ℋ_2(𝐀))=(_ℋ_1∩ℋ_2)^2(_ℋ_1Δℋ_2(𝐀))=_ℋ_1Δℋ_2(𝐀),
    _ℋ_2(𝐀)    ↦    (_ℋ_2)^2(𝐀)=𝐀

since _ℋ_1∩ℋ_2^2=_ℋ_2^2=_n.
So (<ref>) holds for the pair (ℋ_1,ℋ_2)
if and only if it holds for (ℋ_1Δℋ_2,∅).

Take any class 𝒞_q, q∈[P], and two elements of
α,β∈𝒞_q. Then, there exists a path Φ_𝒱(α→β)
in 𝐗(𝒱), which consists of a chain of pairs
(ν_i_T,γ_T), ν_i_T,γ_T∈𝒞_q,
associated with subsets 𝒱_γ_T^ν_i_T∈𝔊.
If the condition (<ref>) holds for such
subsets, one gets 

    ν_i_T∈ℋ_1Δℋ_2⇔γ_T∈ℋ_1Δℋ_2

for all T. The concatenation of all these equivalences for all
the pairs in the path gives the implication 

    (<ref>)⇒(α∈ℋ_1Δℋ_2⇔β∈ℋ_1Δℋ_2)

for all q∈[P] and α,β∈𝒞_q, which is
equivalent to (<ref>).


Thus the redundancy in the representation of signatures by subsets
of [n] is due to the elements of 

    {⋃_q∈𝔭𝒞_q, 𝔭⊆[P]} .

Note that ⋃_q∈𝔭𝒞_q are pairwise
distinct for different choices of 𝔭 since (𝒞_1,…,𝒞_P)
is a partition. This still holds for the elements ℋ_2
in 

    {ℋ_1Δ(⋃_q∈𝔭𝒞_q), 𝔭⊆[P]} .

satisfying (<ref>),
since the symmetric difference is invertible and, hence, the mapping
⋃_q∈𝔭𝒞_q↦ℋ_1Δ(⋃_q∈𝔭𝒞_q)
is bijective. 

 The number of distinct
signatures obtained from sign choices (<ref>)
for columns and (<ref>) for a row is 2^n+1-P,
where P is the number of classes of →_𝒱
associated with 𝐀. 


For each subset ℐ∈𝔊 consider a map constructed
as in (<ref>) that associates ν(α)∈ℐ∖𝒱
to a unique α∈ℐ∖𝒱 so that 𝒱_α^ν(α)∈𝔊.
This also implies α→_𝒱ν(α). Thus,
the non-vanishing minors of 𝐀 correspond to subsets which
intersect all the classes 𝒞_1,…,𝒞_P
and, in particular, 

    #(ℐ∩𝒞_q)=#(𝒱∩𝒞_q)=:k_q,   q∈[P]

does not depend on ℐ∈𝔊. Then, the involutions
Δ_q defined by 

    ℋ↦Δ_q(ℋ):=ℋΔ𝒞_q,  ℋ⊆[n], q∈[P]

act as follows 

    sign[Δ(_Δ_qℋ(𝐀);ℐ)]     =    sign[Δ(_𝒞_q(__ℋ(𝐀));ℐ)]
        =     (-1)^k_q·sign[Δ(__ℋ(𝐀);ℐ)].

Using the partition (𝒞_1,…,𝒞_P), one
can uniquely express ℋ⊆[n] in terms of its components
ℋ_q:=ℋ∩𝒞_q and get, for each
𝔭:={q_1,…,q_T}⊆[P], the following
equalities 

    ℋΔ(⋃_q∈𝔭𝒞_q)     =    (⋃_s∈[P]∖𝔭ℋ_q)∪(⋃_q∈𝔭𝒞_q∖ℋ_q)
        =    Δ_q_1∘Δ_q_2∘⋯∘Δ_q_T(ℋ).

Hence, if ℋ_2=ℋ_1Δ(⋃_q∈𝔭𝒞_q)
(as in (<ref>)),
then from (<ref>) and (<ref>)
one gets 

    sign[Δ(_ℋ_2(𝐀);ℐ)]=(∏_q∈𝔭(-1)^k_q)·sign[Δ(_ℋ_1(𝐀);ℐ)].

Now choose any ancillary soliton parameter κ_0∉{κ_1,…,κ_n},
introduce the matrix 

    :=(1)⊕𝐀=([      1 0⃗_n^T;   0⃗_k      𝐀 ]).

and denote the new column by the index 0. There exists a bijection
between the non-vanishing minors of 𝐀 and those of ,
that is ℐ↦{0}∪ℐ. Any choice of signs
_ℋ for the columns of 𝐀,
ℋ⊆[n], can be extended to a choice for 
as _ℋ:=(1)⊕_ℋ.
Furthermore, the choice R=-1 in (<ref>)
is restated as a new signature for columns only, i.e., (-1)⊕_n=:_{0}.
Hence this equivalent model describes the original one using only
column operations, up to a common multiplicative factor for the terms
Λ_ℐ(x). The pivot set for 
is 𝒱:={0}∪𝒱, and this extends
the relation →_𝒱 to the equivalence →_𝒱
whose classes are 𝒞_1,…,𝒞_P and {0},
since 𝒱_α^0∈𝒫_k[n]∖𝔊
for all α∈[n].

So there are 2^n+1 choices of signs for the columns of 
and P+1 distinct equivalence classes for →_𝒱.
The cardinality of one of them, i.e., #{0}=1, is odd: hence,
from (<ref>), the substitution 𝔭↦{0}∪𝔭
induces the mapping 

    sign[Δ(_ℋ_2();ℐ)]    ↦   sign[Δ(_ℋ_2Δ{0}();ℐ)]
        =     -sign[Δ(_ℋ_2();ℐ)]

for any subset 𝔭⊆[P]. In conclusion, Lemma <ref>
states that, for each ℋ_1⊆{0}∪[n], the
possible sets ℋ_2 satisfying (<ref>)
lie in (<ref>); each
element of this family satisfies (<ref>)
independently on ℐ∈𝔊; finally, from (<ref>)
it follows that each signature 𝔭⊆[P] corresponds
to the opposite one {0}∪𝔭, so they occur in equal
numbers. This means that exactly half of the terms in (<ref>)
have the same signature of ℋ_1, while the other half
have opposite signature. Hence there are 1/22^P+1=2^P
subsets associated with a single signature, and the number of allowed
signatures is 2^n+1/2^P=2^n+1-P. 


 The previous
discussion also implies that the sets (<ref>)
are equipollent and pairwise disjoint, since each of them contains
all the possible combinations of subsets of [n] that induce a given
signature Σ or the opposite -Σ. 

The freedom in the choice of signs generalizes free statistical amoebas
(including an additional row sign flip), which fall within the case
P=1. Indeed, let μ_s denote the number of distinct signatures
induced by the sign flip of exactly s columns of 𝐀,
s∈[n], without limitations on R∈{±1}. Each combination
of signs for row and columns can be labelled by an element in 𝒫_s[n]×{±1},
hence μ_s≤2·ns. At s=n/2,
ℋ∈𝒫_n/2[n] produces the same signature
of its complement [n]∖ℋ∈𝒫_n/2[n]
for an appropriate choice of R, so μ_n/2≤nn/2.
These bounds, along with the action of (<ref>)
and the result in Proposition <ref>,
give 

    2^n=∑_s=0^n/2μ_s=∑_s=0^n/2μ_s+μ_n-s/2≤∑_s=0^nns=2^n.

Thus all the bounds are in fact equalities, i.e., μ_s=2·ns
at s≠n/2 and μ_n/2=nn/2 .



§  LEVELS OF CONSTRAINED
AMOEBAS 


The previous discussion leads to an extension of the concept of statistical
amoeba to higher dimensional cases. Following the construction for
free statistical amoebas in <cit.>, one can focus on the
family of functions (or the associated locus of zeros) obtained from
τ through all the combinations of s sign flips for columns
at s∈[n] fixed.

We can associate with each 𝒮⊆[n] the vector v(𝒮)∈𝔽_2^n
defined as v(α)=1 if α∈𝒮 and v(α)=0
otherwise. The intersection of 𝒮_1,𝒮_2∈𝒫[n]
is given by the componentwise product v(𝒮_1∩ S_2)_α=v(𝒮_1)_α· v(𝒮_2)_α,
hence the parity of #(𝒮_1∩𝒮_2) is equal
to the dot product 

    #(𝒮_1∩𝒮_2)≅∑_α=1^nv(𝒮_1)_α· v(𝒮_2)_α=:v(𝒮_1)· v(𝒮_2)  mod 2.

Motivated by this, we adopt the following notation 

    𝒮∥𝒯   ⇔   #(𝒮∩ 𝒯)≅1  mod 2,
    𝒮⊥𝒯   ⇔   #(𝒮∩ 𝒯)≅0  mod 2

and ℋ⊥𝔏:={ℒ∈𝔏: ℋ⊥ℒ}
with ℋ∈𝒫[n] and 𝔏⊆𝒫[n].

Unlike the free statistical amoeba at k=1, only some strata are
visible under choices of signs allowed by determinantal/integrability
requirements. For instance, a single change of sign preserves neither
the determinantal structure (see Example <ref>)
nor the solution of the KP II equation in generic situations. Hence
the 1-stratum for the free statistical amoeba is not part of the
constrained amoeba.

Let us focus on the case 𝔊=𝒫_k[n]. For all
the choices of _𝒮, s∈[n]
and 𝒮∈𝒫_s[n], the number Ω(n,k;s)
of - signs generated by 𝒮 is equal to the cardinality
of 𝒮∥𝒫_k[n]. This holds for all possible choices of 𝒮∈𝒫_s[n]
by permutation symmetry. The singular locus corresponding to 𝒮
is a subset of all the singular loci in the free Ω(n,k;s)-statistical
amoeba.

If 𝔊=𝒫_k[n], then a choice (<ref>)
with #𝒮=s induces a signature with 

    Ω(n,k;s):=
    {[                ω(n,k;s),                                           m≥ k+s,;         1-(-1)^n-k-s/2nk +(-1)^n-k-sω(n,n-k;n-s),                    m<k+s ].

where 

    ω(n,k;s):=1/2nk-1/2n-sk·_2F_1(-s,-k;n-s-k+1;-1).



Let ℐ∈𝒫_s[n]. The number of subsets 𝒜∈𝒫_k[n]
satisfying 𝒜∥ℐ is 

    ∑_a=0^1/2min{k,s}s2a+1·n-sk-2a+1
        =    ∑_a=0^s(-1)^a+1+1/2·sa·n-sk-a
        =    1/2·∑_a=0^k(-1)^a+1·sa·n-sk-a+1/2·∑_a=0^ssa·n-sk-a

where tw=0 at t<w or w<0. We observe that 𝒜∥𝒮
is equivalent to 

    #(([n]∖𝒮)∩([n]∖𝒜))     =    #[n]∖(𝒮∪𝒜)
       ≅    n-s-k+1  mod 2

where the principle of inclusion-exclusion has been used in the second
line. So, at even n-s-k (respectively, odd n-s-k), the enumeration
of sets 𝒜∈𝒫_k[n] with 𝒜∥𝒮
is equivalent to the enumeration of subsets [n]∖𝒜∈𝒫_n-k[n]
with [n]∖𝒜∥[n]∖𝒮 (respectively,
[n]∖𝒜⊥[n]∖𝒮). Furthermore,
exactly one holds between n-s-k≥0 or n-(n-s)-(n-k)=s+k-n>0. 

So we can look at the situations where n-s-k≥0 and derive the
required quantity in other cases by the previous observation. First
note that 

    ∑_a=0^ssa·n-sk-a=nk.

counts the number of elements of 𝒫_k[n]. For the first
summation, one has 

    ∑_a=0^k(-1)^a+1·sa·n-sk-a
        =     -n-sk·∑_a=0^kka·(-1)^a· s!(n-s-k)!/(s-a)!(n-s-k+a)!
        =     -n-sk·_2F_1(-s,-k;n-s-k+1;-1)

where _2F_1 is the Gaussian hypergeometric function, and all
the equivalences are well-posed due to the condition n-s-k≥0.
Adding the two contributions, we get Ω(n,k;s)=ω(n,k;s).
From this we also find that, at n-s-k<0, the number of (n-k)-subsets
ℬ⊆[n] with ℬ∥([n]∖𝒮)
is ω(n,n-k;n-s), while those with ℬ⊥([n]∖𝒮)
are nk-ω(n,n-k;n-s). By (<ref>)
and subsequent observations, these two quantities respectively enumerate
the number of s-subsets 𝒜⊆[n] with 𝒜∥𝒮
at even n-k-s and at odd n-k-s. These results can be expressed
as in (<ref>), which concludes the proof. 


It is worth remarking that there is a duality between the dimension k and the level s: the number of pairs (ℋ_1;ℋ_2)∈𝒫_k[n]×𝒫_s[n] such that ℋ_1∥ℋ_2 can be enumerated in two ways, i.e., fixing one of the two components ℋ_i, i∈{1,2} and considering all the subsets ℐ with ℋ_i∥ℐ. This double counting implies the following identity   

    ns·Ω(n,k;s)=nk·Ω(n,s;k).

From this, one also finds  

    Ω(n,k;s)<1/2nk⇔ns/nkΩ(n,k;s)<1/2ns⇔Ω(n,s;k)<1/2ns.

In this sense, the distinction between (free) amoebas and antiamoebas, as defined in Section <ref>, is compatible with such a duality. Furthermore, when g_ℐ>0 for all ℐ∈𝒫_k[n], the quantity Ω(n,s;k) dual to Ω(n,k;s) is related to the behaviour of the constrained amoeba at large values of x, namely, to its tropical limit <cit.>: outside the locus where max_ℋ∈𝒫_k[n]Λ_ℋ(x)=:Λ_𝒟(x) is attained more than once, the sign of τ_𝒮(x) at ||x||→∞ coincides with the induced sign for this dominant term, that is (-1)^#(𝒮∩𝒟). So Ω(n,s;k) represents the number of subsets 𝒮∈𝒟∥𝒫_s[n] where τ_𝒮(x)<0.  




§  APPLICATIONS 




 §.§  Application to information
transfer via message coding


The previous results suggest using an initial function (<ref>)
to encode information. Specifically, we can think at each signature
as a message encoded in a G-bits string, where G:=#𝔊.
Here we assume that the order of the bits in the string corresponds
to a given (e.g., lexicographical) order for the elements of 𝒫_k[n].
Analogously, we can represent (<ref>)
as a string with nk entries in {0,1,}, where
the entries label the minors of 𝐀 and the symbol 
is associated with vanishing minors. 

If one knows that the original function τ(x) solves the KP II equation
and receive a new function τ(x)-2·τ_Σ(x),
then the fulfilment of the KP II equation implies that a particular
choice of signs has been sent. We stress that all the signs have to
be checked to confirm that Σ is induced by row/column operations:
indeed, in the generic case G=nk, a single switch of
sign converts a constrained signature to a non-constrained one, as
shown in Example <ref>. However, in the present
approach, one can check if Σ is induced by a choice (<ref>)
indirectly, i.e., without having to find such a configuration
σ_𝒮 and, hence, avoiding the effort to get this
additional knowledge. 

In order to quantify the amount of information that can be acquired
through this single check, we consider the well-known Kullback-Leibler
divergence <cit.>. After the reception of the message,
but before any additional check on the signs, one can recover
the data related to dependence relations among the columns of 𝐀,
namely 𝔊, G and, for any fixed 𝒱∈𝔊, the relation →_𝒱 and the dimensions k_1,…,k_P. At this point, the prior
information is based on strings of bits indexed by 𝔊⊆𝒫_k[n].
With no additional constraint, we can assume a prior distribution u
where all the 2^G G-bits strings have the same statistical weight
2^-G. In the generic case G=nk there are exp(nk·ln2)
such strings, but only 2^n of them satisfy the KP II equation
by Theorem <ref>
and Proposition <ref>. This
restricted family of strings associated with solitonic signatures
is the support for the posterior distribution u_KP. So the Kullback-Leibler
divergence is given by 

    D^⋆_KL(u_KP||u)     :=    ∑_𝒮⊆[n]2^-n·ln(exp(-n·ln2)/exp(-nk·ln2))
        =    ln2·[nk-n].

When #𝔊<nk, the number of distinct general
strings is 2^G, while the number of those satisfying the KP II
equation is 2^n+1-P by Proposition <ref>.
So one gets 

    D^⋆_KL(u_KP||u)=ln2·(G-n+P-1)

which is a real non-negative quantity, hence we also find 

    G+P≥ n+2.


In the previous cases, the assumptions of equiprobability for strings
in {±1}^𝔊 and for choices of signs induced by
(<ref>) are consistent as a result of
Proposition <ref>, since the
equivalence classes associated to the different signatures are equipollent (also see Remark <ref>).
One can get the quantities (<ref>)-(<ref>)
through the choices (R,_𝒮),
R∈{±1} and 𝒮⊆[n], which reduces to the
same multiple-counting of both general and solitonic signatures. This
type of equivalence does not necessarily hold when further restrictions
affect the prior. In particular, we consider a situation where the
number s=#𝒮 for allowed σ_𝒮 is fixed
and known, as in the discussion on statistical amoebas in <cit.>. 

In order to explore some features of this setting, we first estimate
the information content associated with the check of the KP II equation
in the generic situation G=nk. Let us look at the effects
of the knowledge of a fixed value for s at s≠n/2
to avoid the occurrence of solitonic signatures related by the involutions
(<ref>) and, hence, to single out the
contribution of the evidence on s. This enables us to consider
a preliminary check before looking at the KP II equation: if neither
the received string Σ nor its opposite -Σ satisfy
#=Ω(n,k;s), then Σ is not obtained through
(<ref>) with R∈{±1} and, hence, does not satisfy the KP II equation. Furthermore, at Ω(n,k;s)≠1/2nk,
also the row sign R is fixed by this preliminary check, and this
restricts the support of the uniform distributions for the prior u and
the posterior u_KP to 𝒫_Ω(n,k;s)[𝒫_k[n]]
and 𝒫_s[n], respectively. The uncertainty on R still
remains at Ω(n,k;s)=1/2nk . So the Kullback-Leibler
divergence is equal to

    D^s_KL(u_KP||u)=ln[ns^-1·nkΩ(n,k;s)]-ln2·δ_2·Ω(n,k;s),nk.

The last term, which involves the cases Ω(n,k;s)=1/2nk,
does not appear if one also knows that only column operations can
be performed. 

We present a graphical representation of the Kullback-Leibler divergence
at G=nk: the behaviour of (<ref>) at different values of s is shown in Figure <ref>, and can be compared with (<ref>) in Figure <ref>. The analytic continuations of both these formulas have been used.


 
The gain or loss of information can be quantified by the difference
D^s_KL(u_KP||u)-D^⋆_KL(u_KP||u) (see Figure <ref>). It depends on the values n,k,s, as shown in the following example. 

 Let us
consider the generic case 𝔊=𝒫_k[n]. At (n,k,s):=(7,3,1)
we find that G=35, Ω(7,3,1)=15, D_KL^⋆=(35-7)·log2≈19.4081
and D_KL(1)=ln(1/7·3515)≈19.9554,
so D_KL^⋆-D_KL(1)<0. 

On the other hand, moving to the case (n,k,s)=(9,4,4), the quantities
G=126, Ω(9,4,4)=60, D_KL^⋆=(126-9)·log2≈81.0982
and D_KL(4)=ln(94^-1·12660)≈79.7126,
so D_KL^⋆-D_KL(4)>0. 

In contrast to the generic case, situations where 𝔊≠𝒫_k[n]
may exhibit different combinatorial features, and the equivalence
between solitonic signatures (<ref>) and the
induced representations (<ref>) is weakened.
In particular, the constraint on the number of negative signs #
in Σ does not correspond to a fixed number of signs s for
𝒮 in (<ref>), and vice versa.
Furthermore, if one assumes a uniform prior distributions on 𝒫_s[n]
in the construction of solitonic signatures by the sender, then it
is natural to assume that the solitonic signatures are not equiprobable.
In fact, moving from signatures in {±1}^𝔊 to elements
of 𝒫_s[n], the assumption that choices for 𝒮∈𝒫_s[n]
occur with the same probability ns^-1 endows the associated
G-bits solitonic strings with multiplicity weights. Each weight
expresses the redundancy of the representation (<ref>)
counting the number of subsets 𝒮∈𝒫_s[n]
that induce the same signature. At fixed s, these weights do not
coincide for all the signatures in general. For instance, if there
is no subset 𝔭⊆[P] such that 

    ∑_q∈𝔭k_q=2∑_q∈𝔭s_q,

then each involution (<ref>) makes the
index s of the signature change. On the other hand, if (<ref>)
is satisfied, then both 𝒮 and 𝒮Δ⋃_q∈𝔭ℋ_q
have s negative signs and induce the same signature. We clarify
these issues through the following examples.

 Consider the matrix 

    𝐏_1:=
    ([ 1 1 1 0 0 0 0; 0 0 0 1 1 0 0; 0 0 0 0 0 1 1 ]).


The non-vanishing minors for 𝐏_1 can be indexed by elements
in {1,2,3}×{4,5}×{6,7}, so G=12, while it is
easily seen that P=3. Not all the choices in 𝒫_2[7]×{±1}
correspond to distinct signatures, due to coincidences from (<ref>)
and R∈{±1}. In fact, there are 22 distinct solitonic signatures
associated with as many classes in a partition of 𝒫_2[n],
where each class includes all the subsets 𝒮 that returns
the same signature. Assuming a uniform distribution on 𝒫_2[7]×{±1},
we get the following induced distribution for solitonic signatures

    {1/42,1/42,1/42,1/42,1/21,1/21,1/21,1/21,1/42,1/42,.
          .1/21,1/21,1/21,1/21,1/21,1/21,1/21,1/21,1/21,1/21,2/21,2/21} .

The Kullback-Leibler divergence from the uniform prior over 2^𝔊
to (<ref>)
is D_KL(𝐏_1;s=2)≈5.30625. Note that
the relative entropy, starting from 𝐏_1 but without
the information on s=2, is 7·ln2≈4.85203<D_KL(𝐏_1;s=2)
by (<ref>). On
the other hand, using (<ref>)
and (<ref>),
we find that D_KL,unconstrained≈19.4081 and D_KL,s=2≈18.8568
in the generic case. Thus, while the unconstrained case is favourable
in the generic case, the unconstrained one may be preferred when 𝔊≠𝒫_k[n]. 

 Consider the matrix 

    𝐏_2:=
    ([ 1 1 1 1 0 0 0 0 0 0; 1 2 3 4 0 0 0 0 0 0; 0 0 0 0 1 1 1 0 0 0; 0 0 0 0 1 2 3 0 0 0; 0 0 0 0 0 0 0 1 1 1; 0 0 0 0 0 0 0 1 2 3 ]).

It is easily seen that 𝐏_1 is a rectangular block matrix
whose three blocks are full-rank Vandermonde matrices, call them 𝐤_1,𝐤_2,𝐤_3.
Thus one can enumerate the non-vanishing minors of 𝐏_1
as in the previous example and find G=42·32^2=54. 

First we note that the constraints on #𝒮 in (<ref>)
and on the negative signs in Σ do not match: at s=2, the
choice 𝒮={1,2} returns a signature Σ with 36
negative signs, while 𝒮={1,5} gives 27 negative
signs. The operation (<ref>) preserves the
parity #, since G is even, so the choices {1,2}
and {1,5} still have different parity when the choice for R
is included. Vice versa, a fixed number of negative signs for solitonic
signatures does not correspond to the same constraint for #𝒮:
both 𝒮_1={1,5} and 𝒮_2={1,5,9} generate
signatures with 27 negative signs, despite having different cardinalities. 

Let us focus on the case s=2. Before undertaking any check on the
signature, we choose a uniform prior on the 2^G=2^54 possible
ones. In contrast to Example <ref>, distinct
choices of R produce distinct signatures: indeed, two subsets 𝒮_1,𝒮_2∈𝒫_s[n]
inducing opposite signatures Σ_1=-Σ_2 has to be related
by the action of some involutions (<ref>)
respecting (<ref>). But
any operation of this type returns the same signature, since the ranks
of the matrices 𝐤_1,𝐤_2,𝐤_3 are
all even. However, also in this case the uniform distribution on 𝒫_2[n]
induces a non-uniform distribution on the solitonic signatures: for
instance, the signature associated with {8,9} does not coincides
with other ones, while {1,2} and {3,4} lie in the same equivalence
class, and this results in different weights. Computing the relative
entropy as in the previous example, one gets D_KL(𝐏_2;s=2)≈33.0226,
while the unconstrained case gives 46·ln(2). 

The features of non-generic cases and the dependence on the parameters
(n,k,s) can be used to adapt the amount of information content
in the two cases of unconstrained and fixed cardinality for 𝒮
in (<ref>). A specific analysis in this
regard will be carried on in a separate work. 



 §.§  Intersection property and its geometric interpretation


The introduction of the families  

    𝔑(x):={𝒮∈𝒫[n]: τ_𝒮(x)<0} ,  𝔑_s:=𝔑∩𝒫_s[n]

plays a significant role in the identification of the stratified structure of statistical amoebas at k=1. This comes from a simple combinatorial property (see Proposition 5 in <cit.>), which can be extended to the case k>1 as follows. 
 

 If Δ_𝐀(ℐ)·Δ_𝐊(ℐ)≥0
for all ℐ∈𝒫_k[n], then there are no 2k
pairwise disjoint sets in 𝔑(x). 


Let us suppose that such 2k sets exist, i.e., {ℐ_1,…ℐ_2k}⊆𝔑(x)
such that ℐ_a∩ℐ_b=∅ for all
a≠ b. By definition, τ_ℐ_a(x)<0
is equivalent to the inequalities 

    ∑_ℋ∥ℐ_a^ℋ∈𝒫_k[n]Λ_ℋ(x)>∑_𝒦⊥ℐ_a^𝒦∈𝒫_k[n]Λ_𝒦(x)   a∈[2k].

Let us divide these 2k subsets in two classes 𝔉_1:={ℐ_a: 1≤ a≤ k}
and 𝔉_2:={ℐ_a: k+1≤ a≤2k}. Adding
the inequalities (<ref>) for 𝔉_1
term by term, one gets 

    ∑_w=0^⌊k/2⌋(k-2w)·∑_#(ℋ∥𝔉_1)=k-w^ℋ∈𝒫_k[n]Λ_ℋ(x)>∑_w=0^⌊k/2⌋(k-2w)·∑_#(𝒦⊥𝔉_1)=k-w^𝒦∈𝒫_k[n]Λ_𝒦(x).

Similarly, for 𝔉_2 one has 

    ∑_w=0^⌊k/2⌋(k-2w)·∑_#(ℋ∥𝔉_2)=k-w^ℋ∈𝒫_k[n]Λ_ℋ(x)>∑_w=0^⌊k/2⌋(k-2w)·∑_#(𝒦⊥𝔉_2)=k-w^𝒦∈𝒫_k[n]Λ_𝒦(x).

If Λ_ℋ(x)≠0 appears in the left hand
side of (<ref>) (respectively (<ref>)),
then it has non-empty intersection with more than ⌊k/2⌋
elements in 𝔉_1 (respectively, 𝔉_2).
In the same way, if ℋ∥𝔉_2={ℐ_b, b∈[k-w]}
with w<k/2 and distinct ℐ_b, then w_1:=#(ℋ∥𝔉_1)≤ w<k/2.
So #(ℋ⊥𝔉_1)=k-w_1 and
the coefficient of Λ_ℋ(x) in the right
hand side of (<ref>) is k-2w_1≥ k-2w.
Since Λ_ℋ(x)≥0 by hypothesis, this
means that the right hand side of (<ref>) is
an upper bound for the left hand side of (<ref>),
which implies 

    ∑_w=0^⌊k/2⌋(k-2w)·∑_#(ℋ∥𝔉_1)=k-w^ℋ∈𝒫_k[n]Λ_ℋ(x)>∑_w=0^⌊k/2⌋(k-2w)·∑_#(ℋ∥𝔉_2)=k-w^ℋ∈𝒫_k[n]Λ_ℋ(x)

Arguing in the same way for (<ref>), one gets

    ∑_w=0^⌊k/2⌋(k-2w)·∑_#(𝒦∥𝔉_2)=k-w^ℋ∈𝒫_k[n]Λ_ℋ(x)>∑_w=0^⌊k/2⌋(k-2w)·∑_#(ℋ∥𝔉_1)=k-w^ℋ∈𝒫_k[n]Λ_ℋ(x)

which is incompatible with (<ref>), i.e., a contradiction. 


The previous bound also holds in the restriction from 𝔑(x) to each individual family 𝔑_s(x), and this provides an extension of a geometric property that can be stressed in the case s=1 to higher levels s>1. At this purpose, we introduce the matrices 

    _α:=e⃗_α·e⃗_α^T=(δ_βγ·δ_αβ)_β,γ∈[n],  α∈[n]

where {e⃗_α: α∈[n]} is the standard
basis for ℝ^n, and 

    𝐋:=(𝐀·(x)·𝐊)^-1·𝐀·(x).

Note that (𝐀·(x)·𝐊)^-1
exists at (𝐀·(x)·𝐊)≠0
(i.e., outside the singular locus), and 𝐋 is a left-inverse
of 𝐊. In particular, (𝐊·𝐋)^2=𝐊·(𝐋·𝐊)·𝐋=𝐊·_k·𝐋=𝐊·𝐋,
so 𝐊·𝐋 is idempotent. Then, the role played
by α∈[n] in the behaviour of (<ref>)
can be assessed by the sign of 

    τ_{α}(x)/τ(x)    =    ((𝐀·(x)·𝐊)^-1)·(𝐀·_α·(x)·𝐊)
        =    ((𝐀·(x)·𝐊)^-1·𝐀·(x)·(_n-2·_α)·𝐊)
        =    (_k-2𝐋·_α·𝐊)

Taking into account that _α^2=_α,
one can apply Sylvester's determinant identity <cit.>
twice and get 

    (_k-2𝐋·_α·𝐊)     =    (_n-2·_α·𝐊·𝐋)
        =    (_n-2·_α^2·𝐊·𝐋)
        =    (_n-2·_α·𝐊·𝐋·_α).

Note that _α·𝐊·𝐋·_α=⟨𝐊_α|𝐋^α⟩·e⃗_α·e⃗_α^T,
where 𝐊_α (respectively 𝐋^α)
is the vector corresponding to the αth row of 𝐊
(respectively column of 𝐋) and ⟨ | ⟩
is the usual Euclidean scalar product. From the matrix determinant
lemma <cit.>, one finds 

    (_n-2·_α·𝐊·𝐋·_α)=1-2·⟨𝐊_α|𝐋^α⟩·(e⃗_α·e⃗_α^T)=1-2⟨𝐊_α|𝐋^α⟩.

So τ_{α}(x)/τ(x)<0 if
and only if e⃗_α^T·(𝐊𝐋e⃗_α)=⟨𝐊_α|𝐋^α⟩>1/2.
Thus, the result in Proposition <ref>
is equivalent to the property that there exist at most 2k-1 diagonal
entries of 𝐊·𝐋 such that ⟨𝐊_α|𝐋^α⟩>1/2,
independently on n. 



§  DISCUSSION AND FUTURE
PERSPECTIVES


In this work we investigated the effects of particular requirements
connected to a type of complexity reduction, namely determinantal
and integrability constraints, on real-valued partition functions.
Such a reduction can be observed through the statistical amoeba associated
with an initial sum of exponentials (<ref>)
fulfilling the given constraints. In such a framework, the family of allowed choices of signs for pre-exponential terms (<ref>) coincides with the signs induced by row/column sign choices for the coefficient matrix 𝐀. In particular, the consistency with
the KP II equation returns τ-functions for the whole KP hierarchy.
This led to the exploration of the number of distinct signatures for a general 𝐀, levels of constrained statistical amoebas, and their applications in the information-theoretic and geometric settings.

These results give rise to questions on further links between the
combinatorics of complex structures and integrability, some of which
have already been pointed out. In particular, it is worth exploring
in more detail the redundancy in the description of signatures (<ref>)
through subsets (<ref>) and the concept
of instability domains. These issues are also related to the investigation of the tropical limit 
of constrained statistical amoebas, as briefly mentioned in Section <ref>. In fact, these matrix models provide a natural framework for the realization of different tropical concepts, in particular for the nested tropical expansion and the tropical symmetry introduced in <cit.>. The nested expansion relies on the extension of the ordering of individual phases φ_α(x) at points where ||x||→∞ to an ordering of  collective phases ∑_α∈ℐφ_α, ℐ∈𝒫_k[n], while the tropical copies of elements of [n] can be represented as copies of columns of 𝐀. Some additional remarks in this regard are given in Appendix <ref>. 
A careful analysis of these subjects could be useful in the development of concrete models for the thermodynamic and statistical systems mentioned in <cit.>, hence it fits within the increasing number of applications of tropical techniques in the description of physical systems (see, e.g., <cit.>).

Besides the theoretical interest, the previous points prompt a search for new applications of soliton-like structures to the propagation of information. Furthermore, the concept of dimensionality reduction can be studied in more depth in the context of subspaces classification. Indeed,
the bounds discussed in Section <ref> 
may be implemented for the purpose of statistical regression <cit.>,
in particular when generalized/weighted least square methods are employed
(see, e.g., <cit.>). These structures involve bounds for diagonal
elements of a projection matrix (also called leverages for orthogonal
projections) and might be applied in signal processing and machine
learning <cit.>. More generally, the presentation
of statistical amoebas as families of partitions of the type (<ref>)
could be combined with cross-validation techniques. The links between these physical and information-theoretic concepts deserve further investigations for a better understanding, and they will be explored in more detail in a separate paper.


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§  PRESERVATION OF DETERMINANTAL CONSTRAINTS



 There exist sets 𝔳⊆[n],
R(𝔳)⊆[k], and a∈ℝ_⋆^(k-#R(𝔳))×(n-#𝔳) that preserve the Cauchy-Binet expansion (<ref>) up to a factor independent on ℐ∈𝒫_k[n],
namely 

    τ(x)=C·exp(∑_ν∈𝔳φ_ν(x))·(a·Θ(x)_[n]∖𝔳;[n]∖𝔳·K_[n]∖𝔳;[k]∖ R(𝔳))

with C constant. 


Given A∈ℝ_⋆^k× n, let A_0
be the reduced row-echelon form of A with D·A_0=A,
D∈ GL_k(ℝ). Introduce the set 

    𝔳:=⋂_ℐ∈𝔊ℐ={α∈[n]: (α∉ℐ⇒ℐ∈𝒫_k[n]∖𝔊)} .

In particular, from 𝒱∈𝔊
one finds 𝔳⊆𝒱 , so we can consider
R(𝔳):={i∈[k]: ν_i∈𝔳}. The dependence
of 𝔳 and R(𝔳) on A will
be implicit when no ambiguity arises. 

Let π be the permutation of [n] such that both the restrictions
.π^-1|_𝔳 and .π^-1|_[n]∖𝔳 are
increasingly monotone, and π^-1(α)<π^-1(β) for each α∈𝔳, β∈[n]∖𝔳. Similarly, take the permutation ϖ
of [k] such that .ϖ^-1|_R(𝔳) and .ϖ^-1|_[k]∖ R(𝔳)
are both order-preserving and, for any i∈ R(𝔳), j∈[k]∖ R(𝔳),
ϖ^-1(i)<ϖ^-1(j). The action of π on A,
Θ(x) and K via the
matrix representation π:=(δ_π(α),β)_α,β∈[n]
is 

    A↦A·π^-1,  Θ(x)↦π·Θ(x)·π^-1,  K↦π·K,

which preserves the product A·Θ(x)·K
. The matrix π·Θ(x)·π^-1
is still diagonal, since π lies in the normalizer
of diagonal matrices, and its entries are the same of Θ(x).
The parity of the number of inversions induced by π on 
ℐ∈𝒫_k[n] is the same for Δ(A·π^-1;ℐ)
and Δ(π·K;ℐ).
So the action of π preserves the terms in the Cauchy-Binet expansion
(<ref>) up to their permutation
denoted by Λ_ℐ(x)↦Λ_π(ℐ)(x).
Hence, we can express g_ℐ using the order given by
π for columns, and an additional relabelling of the rows via
the left action of the matrix ϖ representing
ϖ: 

    g_ℐ    =    (D)·(ϖ)·(A_0_[k]∖ R(𝔳);ℐ∖𝔳)·VdM(κ;𝔳) 
          ·VdM(κ;ℐ∖𝔳)·∏_α∈𝔳∏_β∈ℐ∖𝔳(κ_β-κ_α).

Introduce 

    P_β:=∏_α∈𝔳(κ_β-κ_α),  β∈[n]∖𝔳

which are non-vanishing under the hypothesis of distinct parameters
κ. So the matrix 

    a:=diag(P_ν^-1: ν∈𝒱∖𝔳)·A_0_[k]∖ R(𝔳);[n]∖𝔳·diag(P_α: α∈[n]∖𝔳)

is well-defined. One can reformulate (<ref>)
as 

    g_ℐ=C·Δ_a(ℐ∖𝔳)·VdM(κ;ℐ∖𝔳)

where 

    C:=(D)·(ϖ)·(∏_ν∈𝒱∖𝔳P_ν)·VdM(κ;𝔳).

does not depend on ℐ. Since the soliton parameters
κ are pairwise distinct, all the terms in (<ref>)
are non-vanishing by definition, thus C≠0. All the non-vanishing
terms in (<ref>) have a common
factor C·exp(∑_ν∈𝔳φ_ν(x)),
and the Cauchy-Binet expansion, along with (<ref>)
and the correspondence 

    ℐ↦ℐ∖𝔳,  ℐ∈𝔊,

allows to formulate the τ(x) as in (<ref>). 

The erasure of rows and columns associated with indices ν∈𝔳
in (<ref>) preserves the reduced
row-echelon form: this is inherited by a when the
normalization diag(P_ν^-1:. .ν∈𝒱∖𝔳)
in (<ref>) is taken into account.
The equality (<ref>) also
implies that a has maximal rank and no null columns.
Note that #(ℐ∖𝔳)=k-#𝔳
for all the terms ℐ∈𝔊.
In particular, 𝒱∖𝔳 is still the minimum
element of 𝒫_k([n]∖𝔳)
associated with a non-vanishing minor of a with respect
to the lexicographic order on [n]∖𝔳 induced
by [n]. Accordingly, a signature Σ for τ induces
a signature for the reduced model (<ref>),
which will still be denoted by Σ 

    Σ(ℐ∖𝔳):=sign(C)·Σ(ℐ).
 


As remarked in the Introduction, we focus on Cauchy-Binet
expansions that generate statistical amoebas relative to the exponential
functions in (<ref>), rather
than determinants. Indeed, each function f(x) can
be trivially expressed as the determinant of a diagonal matrix diag(1,…,1,f(x)),
and no constraints arise. Also the number of degrees of freedom, which
is related to the dimensions of the matrices involved in the expansion,
has to be bounded in order to get non-trivial constraints. In fact,
any sum of functions ∑_t=1^Wf_t(x), W∈ℕ,
can be expressed via (<ref>) as

    [((f_1(x) …  f_W(x)) ⊕_ℓ)·((1  …  1)^T⊕_ℓ)].

This leads us to define determinantal choices of signs, i.e. signatures
preserving the determinantal expansion (<ref>),
as follows. 

 We call
a signature Σ a determinantal choice
of signs, acting on a determinant (<ref>),
if there exist a set {κ̃_α: α∈[n]∖𝔳}
and a matrix ã∈ℝ^#(k-𝔳)×#(n-𝔳)
such that Σ is induced by these data through (<ref>),
up to a common scale factor λ(x). Specifically,
this means that 

    Δ_ã(ℐ∖𝔳)·Δ_k̃(ℐ∖𝔳)·exp(∑_α∈ℐ∖𝔳φ̃_α(x))=λ(x)·Σ(ℐ)·Δ_A(ℐ)·Δ_K(ℐ)·exp(∑_α∈ℐφ_α(x)),  ℐ∈𝔊

where φ̃_α(x):=∑_u=1^dκ̃_α^ux_u
and λ(x)≠0. 

This definition only relies on the data provided by
the terms in the expansion (<ref>)
indexed by 𝔊, as it is shown in the next lemma. 


Assuming that the components of κ are pairwise
distinct, the data n-#𝔳, k-#𝔳
and {κ_α: α∈[n]∖𝔳} are
uniquely determined. Furthermore, a determinantal choice of signs
preserves the absolute values of the maximal minors Δ_a(ℐ)
of the matrix a defined in (<ref>),
up to a multiplicative factor independent on ℐ. 


Let us start from A and the associated sets 𝔊
and 𝔳, and take any α∈[n]∖𝔳.
From the lack of null columns, there exist ℐ,𝒥∈𝔊
with α∈ℐ∖𝒥. The exchange property
(<ref>) implies that there exists β∈𝒥∖ℐ
with ℐ,ℐ_β^α∈𝔊. We
are looking at transformations that preserve each exponential exp(∑_α∈ℐ∑_u=1^dκ_α^ux_u)
in (<ref>) with ℐ∈𝔊,
then the ratio of the exponentials corresponding to ℐ
and ℐ_β^α 

    exp(∑_u=1^d(κ_α^u-κ_β^u)· x_u)=exp(∑_γ∈ℐ∑_u=1^dκ_γ^ux_u)^-1·exp(∑_δ∈ℐ_β^α∑_u=1^dκ_δ^ux_u)

is preserved too. In particular, the coefficients κ_α-κ_β
of x_1 and κ_α^2-κ_β^2 of x_2
are left unchanged. The assumption κ_α≠κ_β
for all α≠β implies that both the quantities κ_α-κ_β
and κ_α^2-κ_β^2/κ_α-κ_β=κ_α+κ_β
are well-defined and fixed. This means that we can recover the values
of data κ_α (and κ_β) for all α∈[n]∖𝔳
from the form of the exponential terms. 

Now look at another full-rank matrix A_1∈ℝ^k_1× n_1
without null columns, which generates the data 𝔊_1
as in (<ref>) and 𝔳_1
as in (<ref>), and a vector κ_1∈ℝ^n_1
such that (<ref>) also holds after
the substitution A↦A_1 and κ↦κ_1.
Same as above, for each α∈[n_1]∖𝔳_1
we can find ℐ∈𝔊_1 and β∈[n_1]∖ℐ
such that α∈ℐ and ℐ_β^α∈𝔊_1
and, from the previous observations, we recover the same set {κ̃_α: α∈[n]∖𝔳}={κ̃_α: α∈[n_1]∖𝔳_1}
of distinguishable soliton parameters. This establishes a correspondence
w: 𝔊⟶𝔊_1 defined by ℐ∖𝔳=w(ℐ)∖𝔳_1
for all ℐ∈𝔊. 

For each α∈[n]∖𝔳, we can choose ℐ(α)∈𝔊
with α∈ℐ(α), which exists by the lack of
null columns in A, then α∈ w(ℐ(α))
too. From (<ref>)
and (<ref>), this means that Δ_A_1(w(ℐ(α)))·Δ_K_1(w(ℐ(α)))≠0, which implies that the parameters (κ_1)_ν,
ν∈𝔳_1, and (κ_1)_α
are pairwise distinct. Since this holds for all α∈[n]∖𝔳,
all the components of κ̃ are pairwise
distinct too. So one has Δ_A(ℐ)=0
if and only if Δ_A_1(w(ℐ))=0. These
data induce the same form (<ref>)
with matrices a and a_1 and multiplicative
constants C,C_1≠0. From the previous discussion VdM(κ;ℐ∖𝔳)=VdM(κ_1;w(ℐ)∖𝔳_1)
for all ℐ∈𝔊, so the equalities (<ref>)
imply |Δ_a(ℐ∖𝔳)|=C_1/C|Δ_a_1(ℐ∖𝔳)|.
□ 


 Assuming that
the parameters κ_1,…,κ_n are pairwise distinct,
a choice of signs is determinantal if and only if it is induced by
a choice of sign for the rows and the columns of A
(up to the action of GL_k(ℝ)). 


We fix the gauge given by the GL_k(ℝ)-action setting 
C_1·λ_1(x)/C·λ(x)=1.
By Remark <ref> and Lemma <ref>,
a signature is determinantal only if the absolute values of maximal
minors of a are preserved. If 𝔳≠∅,
then the overall sign(C) in (<ref>)
can be expressed as a choice of sign for a row in R(𝔳),
and the study is reduced to [k]∖ R(𝔳)
and [n]∖𝔳 through the map (<ref>).
Therefore, to simplify the notation, we can focus on the case a∈ℝ_⋆^k× n
with pivot set 𝒱 without loss of generality.

The equality 

    |a_iβ|=|Δ_a(𝒱_β^ν_i)/Δ_a(𝒱)|=|Δ_ã(𝒱_β^ν_i)/Δ_ã(𝒱)|=|ã_iβ|,   i∈[k],β∈[n]\𝒱

means that the absolute values of the entries of a
are fixed too. So, the transformation (<ref>)
is induced by a choice of signs 

    σ: { (i,α)∈[k]×[n]: a_iα≠0}⟶{±1}

for the non-vanishing entries of a, which will be
denoted by σ(a) as well.

We now construct a sequence of operations to label the rows and columns
of σ(a) with signs ϱ: [k]⟶{±1}
and χ: [n]⟶{±1} respectively.
Let 𝒢:=([k],E)
be a graph whose vertices label the rows of a. The
pair (i,j) is an edge if and only if there
exists γ∈[n] such that a_iγ≠0≠ a_jγ, i.e. 

    (i,j)∈ E⇔∃γ∈[n]: Δ_a(𝒱^ν_i_γ)≠0≠Δ_a(𝒱^ν_j_γ).

So fix an arbitrary row h_1∈[k] of a, e.g.
h_1=1. Without loss of generality, we can set ϱ(h_1):=+1.
Define 𝔠_1:={α∈[n]: a_h_1α≠0}
and χ(α):=σ(h_1,α) for all α∈𝔠_1.
Note that, for each i∈[k], all the products σ(h_1,α)·σ(i,α),
α∈𝔠_1 and a_iα≠0, coincide: indeed,
this is trivially true at i=h_1. If i≠ h_1 and (i,α),
(i,β) are such that α,β∈𝔠_1, then
0∉{a_h_1α,a_h_1β} by definition. Hence,
at a_iα≠0≠ a_iβ, the constraint |Δ_σ(a)(𝒱∖{ν_h_1,ν_i}∪{α,β}})|=|Δ_a(𝒱∖{ν_h_1,ν_i}∪{α,β}})|
is equivalent to 

    |σ(h_1,α)·σ(i,β)· a_h_1α· a_iβ-σ(i,α)·σ(h_1,β)· a_iα· a_h_1β| 
        =    |a_h_1α· a_iβ-a_iα· a_h_1β|.

From a_h_1α· a_iβ· a_iα· a_h_1β≠0,
one gets |a_h_1α· a_iβ-a_iα· a_h_1β|≠|a_h_1α· a_iβ+a_iα· a_h_1β|,
thus 

    σ(h_1,α)·σ(i,β)=σ(i,α)·σ(h_1,β)    ⇔   σ(h_1,α)·σ(h_1,β)·σ(h_1,α)·σ(i,β)
        =    σ(h_1,α)·σ(h_1,β)·σ(i,α)·σ(h_1,β)
       ⇔   σ(h_1,β)·σ(i,β)
        =    σ(h_1,α)·σ(i,α).

So, for any (h_1,i)∈ E , the ith row can be labelled with
a sign ϱ(i):=σ(i,α)·χ(α) for some α∈𝔠_1
assuming that 𝔠_1≠∅. Then let 𝔯_1:={i∈[k]∖{h_1}: (i,h_1)∈ E}
and h_2:=min𝔯_1. By the previous observation, one
has a definite sign ϱ(h_2). Let 𝔠_2:={α∈[n]∖𝔠_1: a_h_2α≠0}
and assign χ(α):=σ(h_2,α)·ϱ(h_2)
for any α∈𝔠_2. As before, for each fixed i∈[k],
all the signs σ(h_2,α)·σ(i,α), α∈𝔠_2
and a_iα≠0, coincide. So the sign ϱ_2(i):=σ(i,α)·χ(α),
for any α∈𝔠_2, is well-defined. It may exist
g∈[k]∖{h_1,h_2} such that a_gα≠0 and
a_gβ≠0 for some α∈𝔠_1, β∈𝔠_2.
We can check that ϱ_2(g)=ϱ(g): in fact, the signs
restricted to rows h_1,h_2,g and columns α,β can
be depicted as 

    [                       α           β;         h_1 ϱ(h_1)·χ(α)           0;         h_2           W ϱ(h_2)·χ(β);           g   ϱ(g)·χ(α) ϱ_2(g)·χ(β) ]

If W≠0, then it equals ϱ(h_2)·χ(α) by
definition. Thus, the constraint |Δ_a(ℐ)|=|Δ_σ(a)(ℐ)|
at ℐ=𝒱∖{ν_h_2,ν_g}∪{α,β}
implies that 

    ϱ(h_2)·χ(α)·ϱ_2(g)·χ(β)=ϱ(h_2)·χ(β)·ϱ(g)·χ(α)

that is ϱ_2(g)=ϱ(g). Now assume that W=0. Since
(h_2,h_1)∈ E, there exists γ∈[n] such that a_h_1γ≠0≠ a_h_2γ,
and γ is clearly different from α and β since
a_h_1β=0=a_h_2α, then consider the extended scheme

    [                       α           β           γ;         h_1 ϱ(h_1)·χ(α)           0 ϱ(h_1)·χ(γ);         h_2           0 ϱ(h_2)·χ(β) ϱ(h_2)·χ(γ);           g   ϱ(g)·χ(α) ϱ_2(g)·χ(β)           X ]

If X≠0 (hence |a_gγ|≠0), then X=ϱ(g)·χ(γ) and, applying |Δ_a(ℐ)|=|Δ_σ(a)(ℐ)| at ℐ=𝒱∖{ν_h_2,ν_g}∪{β,γ}, one gets ϱ_2(g)=ϱ(g). Also note that this can be found without evaluating X from the requirement |Δ_a(ℐ)|=|Δ_σ(a)(ℐ)| at both ℐ=𝒱∖{ν_h_1,ν_g}∪{α,γ} 
and ℐ=𝒱∖{ν_h_2,ν_g}∪{β,γ} simultaneously, i.e.  

    +1     =    ϱ(h_1)·χ(α)· X·ϱ(h_1)·χ(γ)·ϱ(g)·χ(α)
        =    ϱ(h_2)·χ(β)· X·ϱ(h_2)·χ(γ)·ϱ_2(g)·χ(β)

that gives 

    ϱ(h_1)·χ(α)·ϱ(h_1)·χ(γ)·ϱ(g)·χ(α)=ϱ(h_2)·χ(β)·ϱ(h_2)·χ(γ)·ϱ_2(g)·χ(β).

If instead X=0, considering the constraint |Δ_a(ℐ)|=|Δ_σ(a)(ℐ)|
in the case ℐ=𝒱∖{ν_h_1,ν_h_2,ν_g}∪{α,β,γ},
we get 

    |-a_h_1γ· a_h_2β· a_gα-a_h_2γ· a_gβ· a_h_1α|
        =     |ϱ(h_1)·χ(γ)·ϱ(h_2)·χ(β)·ϱ(g)·χ(α)· a_h_1γ· a_h_2β· a_gα
        +    ϱ(h_2)·χ(γ)·ϱ_2(g)·χ(β)·ϱ(h_1)·χ(α)· a_h_2γ· a_gβ· a_h_1α|

and, from 0≠ a_h_1γ· a_h_2β· a_gα· a_h_2γ· a_gβ· a_h_1α,
we obtain 

    ϱ(h_1)·χ(γ)·ϱ(h_2)·χ(β)·ϱ(g)·χ(α)=ϱ(h_2)·χ(γ)·ϱ_2(g)·χ(β)·ϱ(h_1)·χ(α).

Since all the factors in (<ref>) and (<ref>)
belong to {±1} and, in particular, they are non-vanishing and
idempotent, these expressions simplify as 

    ϱ_2(g)=ϱ(g).

and the sign ϱ_2(g)·χ(γ)=ϱ(g)·χ(γ)
can be consistently fixed. This construction can be extended through the iterations 

    𝔯_r-1    :=    { i∈[k]∖{h_1,…,h_r-1}: (i,h_r-1)∈ E} ,
    
    h_r    :=    min𝔯_r-1,
    𝔠_r    :=    {α∈[n]∖⋃_t=1^r-1𝔠_t: a_h_rα≠0}

and the signs ϱ_r-1(h_r) and χ(α):=σ(h_r,α)·ϱ_r-1(h_r)
for any α∈𝔠_r. If the signs ϱ_s,
as long as χ, are uniquely defined at steps s<r, then one
can check the compatibility of ϱ_r with ϱ too,
formally ϱ_r(g)=ϱ(g) for all rows g connected
to both h_r and h_s, s<r. At this purpose, it is worth
introducing paths on a, that are defined
as finite sequences of non-vanishing elements of a
connected by alternate moves along rows and columns, i.e. chains of
the type (<ref>). The process that links the signs ϱ_r(g),
derived from h_r, and ϱ(g), obtained in a previous step
h_s, can be represented by a path Φ starting with (g,α_1)→(h_i_2,α_1)→…,
passing through some signed rows Φ_ϱ:={g,h_i_2,…,h_i_L}
and columns Φ_χ:={α_1,…,α_L},
and ending with …→(h_i_L,α_L)→(g,α_L),
where α_1=α_s∈𝔠_s, α_L=α_r∈𝔠_r
and a_gα_s≠0≠ a_gα_r. 

Note that the column of a associated with any α_T∈Φ_χ has at least two non-vanishing entries (h_i_T,α_T) and (h_i_T+1,α_T), while there is only one non-vanishing entry in pivot columns by definition, hence 𝒱∩Φ_χ=∅. Suppose that there
is a row h̅∈Φ_ϱ with 0∉{a_h̅α_T-1,a_h̅α_T,a_h̅α_X},
X∉{T-1,T}, and first assume that T<X: since the column
α_X is reached at (h_i_X,α_X), we can substitute
the chain (h̅,α_T-1)→(h̅,α_T)→…→(h_i_X,α_X)→(h_i_X+1,α_X)
with (h̅,α_T-1)→(h̅,α_X)→(h_i_X+1,α_X)
in the path. At X<T-1, likewise, we can change (h_i_X,α_X)→(h_i_X+1,α_X)→…→(h̅,α_T-1)→(h̅,α_T)
with (h_i_X,α_X)→(h̅,α_X)→(h̅,α_T).
In both cases, the result is a path connecting α_s to α_r
with shorter length, which can still be used to compare ϱ_r(g)
and ϱ(g). Similar substitutions can be carried out for columns c̅∈Φ_χ that appear as components of more than two elements of the path. So we focus on paths of minimal length: the previous
construction shows that the submatrix extracted from a
selecting the rows in Φ_ϱ and columns in Φ_χ
associated with a minimal path Φ from (g,α_s) to (g,α_r) has exactly two non-vanishing
elements per row and column. Hence, the condition |Δ_a(𝒱∖{ν_i: i∈Φ_ϱ}∪Φ_χ)|=|Δ_σ(a)(𝒱∖{ν_i: i∈Φ_ϱ}∪Φ_χ)|
can be expressed as in (<ref>), that is

    ϱ(g)·χ(α_1)·(∏_T=2^Lϱ(h_i_T)·χ(α_T))=(∏_T=1^L-1ϱ(h_i_T+1)·χ(α_T))·ϱ_2(g)·χ(α_L)

which gives ϱ_r(g)=ϱ(g).  

These steps can be repeated while 𝔯_r≠∅≠𝔠_r:
in this process we start from a node h_1 of the graph 𝒢
and follow a path of adjacent vertices. If 𝔯_r=∅,
then we can follow this path backwards until we reach 𝔯_u,
u∈[r-1], such that there exists h̃_u∈𝔯_u
with h̃_u≠ h_w, w∈[r-1]. Then, these operations
can be repeated along another path of adjacent nodes starting from
h̃_u. At each stage an additional sign is selected compatibly with the previous
ones. Such a process explores each node in the connected
component of 𝒢 containing h_1 exactly once. Repeating these steps for all the connected components of 𝒢
we give a sign to all the rows of a. Since each column
α contains at least one non-vanishing element a_h_rα≠0
and all the rows are visited, the index α belongs to 𝔠_r
at certain step r. Hence, all the columns are labelled by a sign
as well. By construction, one has σ(i,α)=ϱ(i)·χ(α)
for all (i,α) such that a_iα≠0. Clearly, this assignment produces a determinantal choice of signs. 


 The proof of Theorem
<ref> is constructive:
it generates one of the possible sign configurations that induce a
given choice Σ. This configuration is not unique, e.g. switching
the sign of two distinct rows of A or a
returns the same Σ. The uniqueness of the previous construction
follows from the choice of a (arbitrary) σ in (<ref>)
for the entries of the reduced row echelon form a
and signs of distinguished nodes (such as h_1) in the connected
components of 𝒢. Furthermore, the algorithm attributes a sign
to vanishing entries of a too. The labelling +0 or
-0 can be thought as the sign of the associated entry in a perturbed
matrix b such that sign(Δ_a(ℐ))=sign(Δ_b(ℐ))
for all ℐ∈𝒫_k[n] with Δ_a(ℐ)≠0.
These points are relevant when a has many vanishing
entries. 




§  REMARKS ON THE TROPICAL LIMIT
 

With regard to the issues dealt with in the present work, tropical
methods have proved useful in the study of algebraic amoebas <cit.>
and the analysis of KdV and KP II soliton solutions and their singularities
<cit.>. Here we briefly discuss
how the models we have described give a concrete realization to the
concepts discussed in <cit.>, especially the role
of order and enumeration in the tropical limit in statistical physics. Relevant
quantities, like the free energy -k_BT·ln𝒵 associated
with (<ref>), fit naturally in this
process, and their tropical limit is suitable for the description
of phenomena like exponential degenerations of energy levels and limiting
temperatures, see <cit.>. 

Determinantal partition functions (<ref>)
give a concrete realization of the nested tropical limit
discussed in <cit.>. For the sake of concreteness, let us consider the case when g_ℐ≥0 for all ℐ∈𝒫_k[n]. Outside the locus of points x∈ℝ^d
where φ_α(x)=φ_β(x)
for some α≠β, one has φ_π(1)(x)>…>φ_π(n)(x)
for a permutation π∈𝒮_n. Here, for large values
of x, there exists exactly one dominant term Λ_𝒟(x)>Λ_ℋ(x),
ℋ∈𝒫_k[n]∖{𝒟}, as follows
from the polynomial form of φ_α(x). In
particular, 𝒟 is the least set in 𝒫_k[n]
associated with a non-vanishing g_𝒟 with respect to
the lexicographical order induced by (π(1),…,π(n)). Hence
one has Λ_ℋ(x)≪Λ_𝒟(x)
for all ℋ≠𝒟 and τ_ℐ(x)<0
if and only 𝒟∥ℐ. Moreover, if g_ℐ>0 for all ℐ∈𝒫_k[n], then the number of such subsets ℐ with odd intersection with 𝒟 is Ω(n,s;k), that is the dual quantity of Ω(n,k;s) as
in (<ref>) and (<ref>).
The nested form (see <cit.>, §6) comes from the identification
of dominant term 𝒟: if one sets 𝔇(r):={ℋ∈𝒫_k[n]: max{i∈[k]∪{0}: {π(1),…,π(i)}⊆ℋ∩𝒟}=r},
then τ can be expressed as 

    τ(x)
        =     e^φ_π(1)(x)(e^φ_π(2)(x)(… e^φ_π(k-1)(x)(e^φ_π(k)(x)(g_𝒟+∑_ℐ∈𝔇(k-1)g_ℐ(x) e^-φ_π(k)(x)∏_α∈ℐ∖π([k-1])e^φ_α(x))...
        +    ..∑_ℐ∈𝔇(k-2)g_ℐ(x) e^-φ_π(k-1)(x)∏_α∈ℐ∖π([k-2])e^φ_α(x))+…∑_ℐ∈𝔇(1)g_ℐ(x)e^-φ_π(2)(x)∏_α∈ℐ∖{π(1)}e^φ_α(x))
        +    .∑_ℐ∈𝔇(0)g_ℐ(x) e^-φ_π(1)(x)∏_α∈ℐe^φ_α(x)).

In such a way, one can recursively find the first k dominant terms,
thus realising a refinement of the tropical limit of the exponents
φ_α through the limit for the collections ∑_α∈ℐφ_α,
ℐ∈𝒫_k[n].

The tropical limit drastically affects the enumeration process too,
mainly due to idempotence. This property can be seen as a symmetry
of the statistical system with respect to creation of copies of its
constituents: if the copying process involves all the constituents
at once, e.g. all the distinct terms in the partition function, the
symmetry is global, otherwise it is local (for more details, see <cit.>,
§8). Matrix models also provide a realization of such a symmetry:
for each α∈[n], one can introduce a “copy” the αth
soliton as a copy of the associated row/column, formally 

    A   ↦   T_α(A):=A·T_α,
    K   ↦   T_α(K):=T_α^T·K,
    Θ(x)    ↦   T_α(Θ(x)):=π_cyc,α^-1·(Θ(x)⊕(e^φ_α(x)))·π_cyc,α

where π_cyc,α is the matrix associated with the cyclic permutation (α+1 α+2 … n+1) and

    (T_α)_γβ:=∑_ω=1^αδ_γ,β·δ_γ,ω+∑_ω=α^nδ_γ,β-1·δ_γ,ω,  α,γ∈[n], β∈[n+1].

If the same number of copies, say ℓ, are created for all α∈[n],
then the resulting τ-function gets a multiplicative factor (ℓ+1)^k
which disappears in the derivation (<ref>),
so the soliton solution is preserved. Thus, there is an intrinsic
tropical global symmetry in this particular class of solutions, which
is also compatible with the GL_k(ℝ)-action defining
the Grassmannian. Further, the effect of the tropical copy of a single
soliton κ_α_0 on the singular locus defined by (A·Θ(x)·K)=0
is the same as its erasure, since 

    (T_α_0^(ℓ)(A)·T_α_0^(ℓ)(Θ(x))·T_α_0^(ℓ)(K))
        =    ℓ·∑_α_0∈ℐ^ℐ∈𝒫_k([N])Δ_ℐ(A)Δ_ℐ(K)exp(∑_β∈ℐφ_β(x))
        =     -ℓ·∑_α_0∉ℐ^ℐ∈𝒫_k([N])Δ_ℐ(A)Δ_ℐ(K)exp(∑_β∈ℐφ_β(x)).

This process is independent on the choice of ℓ>1, since the
factor ℓ in (<ref>)
disappears in the derivation (<ref>). In this
sense, the invariance under tropical copies is local on the singular
locus. 

