On the degree and half-degree principle for symmetric polynomials

Riener, Cordian

On the degree and half-degree principle for symmetric polynomials

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Datum

2012

Publikationstyp

Zeitschriftenartikel

Erschienen in

Journal of Pure and Applied Algebra ; 216 (2012), 4. - S. 850-856. - ISSN 0022-4049

Zusammenfassung

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 R^n 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 S_n 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.

Fachgebiet (DDC)

510 Mathematik

Zitieren

ISO 690

RIENER, Cordian, 2012. On the degree and half-degree principle for symmetric polynomials. In: Journal of Pure and Applied Algebra. 216(4), pp. 850-856. ISSN 0022-4049. Available under: doi: 10.1016/j.jpaa.2011.08.012

BibTex

@article{Riener2012degre-17510,
  year={2012},
  doi={10.1016/j.jpaa.2011.08.012},
  title={On the degree and half-degree principle for symmetric polynomials},
  number={4},
  volume={216},
  issn={0022-4049},
  journal={Journal of Pure and Applied Algebra},
  pages={850--856},
  author={Riener, Cordian}
}

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