Abstract:
The classical Descartes' rule of signs bounds the number of positive real roots of a univariate real polynomial in terms of the number of sign variations of its coefficients. This is an extremely simple rule, which is exact when all the roots are real, for instance, for characteristic polynomials of symmetric matrices. No general multivariate generalization is known for this rule, not even a conjectural one.
I will gently describe two partial multivariate generalizations obtained in collaboration with Stefan Müller, Elisenda Feliu, Georg Regensburger, Anne Shiu, Carsten Conradi and Frédéric Bihan. Our approach shows that the number of positive roots of a polynomial system of n polynomials in n variables is related to the relation between the signs of the maximal minors of the matrix of exponents and of the matrix of coefficients (that is, to the relation between the associated oriented matroids). I will explain which are the main challenges to devise a complete multivariate generalization.