Optimal Descartes rule of signs for polynomial systems supported on circuits

ALICIA DICKENSTEIN

Berlín

Congreso; Complexity of numerical computation; 2019

Descartes rule of signs for univariate real polynomials is a beautifullysimple upper bound for the number of positive real roots. Moreover, it givesthe exact number of positive real roots when the polynomial is real rooted, forinstance, for characteristic polynomials of symmetric matrices. A generalmultivariate Descartes rule is certainly more complex and still elusive. I will recall the few known multivariate cases and will present a new optimalDescartes rule for polynomials supported on circuits, obtained in collaborationwith Frédéric Bihan and Jens Forsgård.