Maximally positive polynomial systems supported on circuits

Abstract

A real polynomial system with support W ⊂ Z n is called maximally positive if all its complex solutions are positive solutions. A support W having n + 2 elements is called a circuit. We previously showed that the number of non-degenerate positive solutions of a system supported on a circuit W ⊂ Z n is at most m(W) + 1, where m(W) ≤ n is the degeneracy index of W. We prove that if a circuit W ⊂ Z n supports a maximally positive system with the maximal number m(W) + 1 of non-degenerate positive solutions, then it is unique up to the obvious action of the group of invertible integer affine transformations of Z n . In the general case, we prove that any maximally positive system supported on a circuit can be obtained from another one having the maximal number of positive solutions by means of some elementary transformations. As a consequence, we get for each n and up to the above action a finite list of circuits W ⊂ Z n which can support maximally positive polynomial systems. We observe that the coefficients of the primitive affine relation of such circuit have absolute value 1 or 2 and make a conjecture in the general case for supports of maximally positive systems.

Sparse Polynomial Systems with many Positive Solutions from Bipartite Simplicial Complexes

“…In order to illustrate this result, we check that some classical families of polytopes, namely order polytopes, hypersimplices, cross polytopes and alcoved polytopes, admit regular unimodular balanced triangulations and thus provide point configurations supporting maximally positive systems. As a by-product, we verify that these classes of point configurations have a basis of affine relations with coefficients in {−2, −1, 1, 2}: this gives evidence in favor of Bihan's conjecture [3,Conjecture 0.6].…”

“…Viro's method [44] (see also [3,33,42] for instance) is one of the roots of tropical geometry and has been used with great success for constructing real algebraic varieties with interesting topological types. It allows to recover under certain conditions the topological type for t close to 0 of a real algebraic variety defined by a system whose coefficients depend polynomially on a positive parameter t. Here we apply a version of Viro's method which has already been used in [40].…”

“…One goal of the present paper was to analyze under which conditions this construction produces polynomial systems whose all toric complex solutions are positive. Such polynomial systems are called maximally positive in [3], where a conjecture about their supports has been proposed [3,Conjecture 0.6].…”

Descartes’ Rule of Signs for Polynomial Systems Supported on Circuits

“…Then, the maximum possible bound (2.13) in Theorem 2.9 for a coefficient matrix C and support A is equal to 5 = vol Z (A) = vol ZA (A) = n + 1. However, Theorem 0.2 in [2] shows that there cannot be vol Z (A) positive real solutions to this system, so the bound in (2.13) is not sharp in this case.…”

“…In [2], polynomial systems supported on a circuit in R n and having n+1 positive solutions have been obtained with the help of real dessins d'enfants. In [12], the main tool for constructing such systems is the generalization of the Viro's patchworking theorem obtained in [16].…”