On the degree and half-degree principle for symmetric polynomials

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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.

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ISO 690RIENER, Cordian, 2012. On the degree and half-degree principle for symmetric polynomials. In: Journal of Pure and Applied Algebra. 2012, 216(4), pp. 850-856. ISSN 0022-4049. Available under: doi: 10.1016/j.jpaa.2011.08.012
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@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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