## Section: New Results

### Construction of sparse polynomial systems with many positive solutions

Participant : Pierre-Jean Spaenlehauer [contact] .

In collaboration with Frédéric Bihan (Univ. Savoie Mont-Blanc), we propose a variant of the classical Viro method to construct polynomial systems with prescribed monomial support and many solutions whose coordinates are all positive [19] . This is an asymptotic construction which has strong connections with tropical and convex geometry, and which involves computational problems such as low-rank matrix completion.