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“Towards a multidimensional Descartes
rule (but still far away)”
Date: 
Downloadfiles: 
Time: 
Wednesday,
05. April 2017 
VideoRecording for any system with MP4support

Video.mp4 (ca.386 Mb) 
15:15 – 16:15 
Alicia
Dickenstein
(Universidad de
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:
http://agenda.albanova.se/categoryDisplay.py?categId=301
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