Optimal Descartes rule of signs for polynomial systems supported on circuits

A DICKENSTEIN

Conferencia; Latin American Geometría Algebraica Real y Tropical Seminar; 2021

Descartesrule of signs for univariate real polynomials is a beautifully simple upperbound for the number of positive real roots. Moreover, it gives the exactnumber of positive real roots when the polynomial is real rooted, for instance,for characteristic polynomials of symmetric matrices. A general multivariateDescartes rule is certainly more complex and still elusive.  I will recallthe few known multivariate cases and will present a new optimal Descartes rulefor polynomials supported on circuits, obtained in collaboration with FrédéricBihan and Jens Forsgård. If time permits, I will talk a bit about lower bounds.