Section: New Results

A polyhedral method for sparse systems with many positive solutions

Participant : Pierre-Jean Spaenlehauer.

Together with Frédéric Bihan (Université Savoie Mont Blanc) and Francisco Santos (Universidad de Cantabria), we investigated in [4] a version of Viro's method for constructing polynomial systems with many positive solutions, based on regular triangulations of the Newton polytope of the system. The number of positive solutions obtained with our method is governed by the size of the largest positively decorable subcomplex of the triangulation. Here, positive decorability is a property that we introduced and which is dual to being a subcomplex of some regular triangulation. Using this duality, we produced large positively decorable subcomplexes of the boundary complexes of cyclic polytopes. As a byproduct we obtained new lower bounds, some of them being the best currently known, for the maximal number of positive solutions of polynomial systems with prescribed numbers of monomials and variables. We also studied the asymptotics of these numbers and observed a log-concavity property.