REV - BASIC NOTIONS SEMINAR at ICTP (Alicia DICKENSTEIN, Simons Senior Research Associate, ICTP) - Thursday 16 November, at 16:00 hrs.

[ICTP Math Section]

BASIC NOTIONS SEMINAR SERIES   2017


Thursday 16 November, at 16:00 hrs.



Alicia Dickenstein (University of Buenos Aires, Argentina)


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


*REVISED 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.



VENUE:  Budinich Lecture Hall (ICTP Leonardo da Vinci Building)