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On the degree and half-degree principle for symmetric polynomials

7 years 9 months ago
On the degree and half-degree principle for symmetric polynomials
In this note we aim to give a new, elementary proof of a statement that was first proved by Timofte (2003) [15]. It says that a symmetric real polynomial F of degree d in n variables is positive on Rn (or on View the MathML source) if and only if it is non-negative on the subset of points with at most max{⌊d/2⌋,2} distinct components. We deduce Timofte’s original statement as a corollary of a slightly more general statement on symmetric optimization problems. The idea that we are using to prove this statement is that of relating it to a linear optimization problem in the orbit space. The fact that for the case of the symmetric group Sn this can be viewed as a question on normalized univariate real polynomials with only real roots allows us to conclude the theorems in a very elementary way. We hope that the methods presented here will make it possible to derive similar statements also in the case of other groups.
Cordian Riener
Added 19 Jun 2012
Updated 19 Jun 2012
Type Journal
Year 2012
Where Journal of Pure and Applied Algebra
Authors Cordian Riener
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