Towards a multivariate Descartes rule (but still far away)

ALICIA DICKENSTEIN

Río de Janeiro

Congreso; Coloquio Brasileiro de Matemática; 2019

The classicalDescartes' rule of signs bounds the number of positive real roots of aunivariate real polynomial in terms of the number of sign variations of itscoefficients. This is an extremely simple rule, which is exact when all theroots are real, for instance, forcharacteristic polynomials of symmetric matrices. No generalmultivariate generalization is known for this rule, not even a conjectural one.I will gently describe two partial multivariate generalizationsobtained in collaboration with Stefan Müller, Elisenda Feliu, GeorgRegensburger, Anne Shiu, Carsten Conradi and Frédéric Bihan. Our approach showsthat the number of positive roots of a  polynomial system of n polynomialsin n variables is related to the  relation between the signs of themaximal minors of the matrix of exponents and of the matrix of coefficients(that is, to the relation between the associated oriented matroids). I willexplain which are the main challenges to devise a complete multivariategeneralization.