Vorträge im Schwerpunkt Reelle Geometrie und Algebra
Freitag, 30. April 2010, um 14:15 Uhr in F426
(Oberseminar)
Cordian Riener
(Frankfurt)
Positivity of symmetric
polynomials
The question of certifying that a given polynomial
in n
real variables is positive has been one of the main motivations for the
development of modern real algebraic geometry in the beginning 20th
century.
In this talk we discuss the question in the context of symmetric
polynomials. Especially we will focus on an elementary proof of a
statement concerning positivity in the symmetric setting that
was first noted by Vlad Timofte. It says that a symmetric real
polynomial F
of degree d
in n
variables is positive on ℝn (
on ℝn+) if
and only if it is so on the subset of points with at
most max{⌊d/2⌋,2} distinct components.
The key idea of the new proof lies in the representation of the orbit
space. The fact that for the case of the symmetric group Sn
it can be viewed as the set of normalized univariate real polynomials
with only real roots allows us to conclude the theorems in a very
elementary way.
zuletzt
geändert am 23. April 2010
