Alicia Dickenstein: Towards a multidimensional Descartes rule (but still far away)

Schedule

14:00-15:00 Prelcolloquium in Room FB42. Speaker TBA

15:15-16:15 Colloquium lecture by Alicia Dickenstein (Room Oskar Klein, AlbaNova)

16:15-17:00 SMC social get together with refreshments 

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 square
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 present an application of our results in the realm of
biochemical reaction networks and will explain which are the main
challenges to devise a complete multivariate generalization.