Optimal Descartes rule of signs for polynomial systems supported on circuits

A DICKENSTEIN

Conferencia; Nonlinear Algebra Seminar Online; 2021

Max Planck Institute for Mathematics in the Sciences, Leipzig

Descartes rule of signs for univariate real polynomials is a beautifully simple upper bound for the number of positive real roots. Moreover, it gives the exact number of positive real roots when the polynomial is real rooted, for instance, for characteristic polynomials of symmetric matrices. A general multivariate Descartes rule is certainly more complex and still elusive.  I will recall the few known multivariate cases and will present a new optimal Descartes rule for polynomials supported on circuits, obtained in collaboration with Frédéric Bihan and Jens Forsgård.