Towards a multidimensional Descartes rule (but still far away)

A DICKENSTEIN

Conferencia; ICTP Basic Notions Seminar; 2017

ICTP

Theclassical Descartes' rule of signs bounds the number of positive real roots ofa univariate 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, for characteristic polynomials of symmetricmatrices. No general multivariate generalization is known for this rule, noteven a conjectural one.Iwill gently describe two partial multivariate generalizations obtained incollaboration with Stefan Müller, Elisenda Feliu, Georg Regensburger, AnneShiu, Carsten Conradi and Frédéric Bihan. Our approach shows that the number ofpositive roots of a polynomial system of n polynomials in n variables isrelated to the relation between the signs of the maximal minors of the matrixof exponents and of the matrix of coefficients (that is, to the relationbetween the associated oriented matroids). I will explain which are the mainchallenges to devise a complete multivariate generalization.