Characterization of circuits supporting polynomial systems with the maximal number of positive solutions

Abstract

A polynomial system with n equations in n variables supported on a set W ⊂ R n of n + 2 points has at most n + 1 non-degenerate positive solutions. Moreover, if this bound is reached, then W is minimally affinely dependent, in other words, it is a circuit in R n . For any positive integer number n, we determine all circuits W ⊂ R n which can support a polynomial system with n+1 non-degenerate positive solutions. Restrictions on such circuits W are obtained using Grothendieck's real dessins d'enfant, while polynomial systems with n + 1 non-degenerate positive solutions are constructed using Viro's combinatorial patchworking.

Descartes’ Rule of Signs for Polynomial Systems Supported on Circuits

“…Theorem 3.3 gives a necessary condition on a circuit A ⊂ Z n for the existence of a coefficient matrix C such that n A (C) reaches the maximal possible value n + 1. Recently, Boulos El Hilany [6] has obtained a necessary and sufficient condition on A for this to hold. The first author of the present paper has obtained a partial generalization of Descartes' rule of signs for any polynomial system.…”