Section: New Results

Constructing Sparse Polynomial Systems with Many Positive Solutions

Participant : Pierre-Jean Spaenlehauer [contact] .

This is a joint work with Frédéric Bihan (Univ. de Savoie, LAMA). Most of the results have been obtained in 2015 [25]; we improved the results during 2016.

Consider a regular triangulation of the convex-hull $P$ of a set $𝒜$ of $n$ points in ${ℝ}^d$, and a real matrix $C$ of size $d\times n$. A version of Viro's method allows to construct from these data an unmixed polynomial system with support $𝒜$ and coefficient matrix $C$ whose number of positive solutions is bounded from below by the number of $d$-simplices which are positively decorated by $C$ (a $d$-simplex is positively decorated by $C$ if the $d\times (d+1)$ sub-matrix of $C$ corresponding to the simplex has a kernel vector all coefficients of which are positive). We show that all the $d$-simplices of a triangulation can be positively decorated if and only if the triangulation is balanced, which in turn is equivalent to the fact that its dual graph is bipartite. This allows us to identify, among classical families, monomial supports which admit maximally positive systems, giving some evidence in favor of a conjecture due to Bihan. We also use this technique in order to construct fewnomial systems with many positive solutions.