Towards a multidimensional Descartes rule (but still far away)

ALICIA DICKENSTEIN

Conferencia; Cattedra Fubini; 2019

Politécnico de Turín

The classical Descartes´ ruleof signs bounds the number of positive real roots of a univariate realpolynomial in terms of the number of sign variations of its coefficients. Thisis an extremely simple rule which is exact when all the roots are real, forinstance, for characteristic polynomials of symmetric matrices. No generalmultivariate generalization is known for this rule, not even a conjectural one. I will gently describe twopartial multivariate generalizations obtained in collaboration with StefanMüller, Elisenda Feliu, Georg Regensburger, Anne Shiu, Carsten Conradi, Frédéric Bihan and Jens Forsgaard. Our approach shows that the number ofpositive roots of a square polynomial system (of n polynomials in n variables)is related to the signs of the maximal minors of the matrix of exponents and ofthe matrix of coefficients. I will also explain which are the main challengesto devise a complete multivariate generalization.