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“Towards a multidimensional Descartes rule (but still far away)”

Date:

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Time:

Wednesday, 05. April 2017

Video-Recording for any system with MP4-support

- Video.mp4 (ca.386 Mb)

15:15 – 16:15

Alicia Dickenstein

(Universidad de Buenos Aires, Argentina)

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.

The future colloquium program can be found at: