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arXiv · 2606.27551

Any-dimensional Positivstellens\"atze for symmetric functions

Sebastian Debus·Robin Schabert
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Abstract

Positivstellens\"atze provide certificates of positivity for polynomials. Extending these certificates to symmetric functions, uniformly across all dimensions, presents structural challenges. For instance, the underlying domain is not semialgebraic. In this paper, we prove two Positivstellens\"atze for symmetric functions that are uniformly bounded below by some $\varepsilon > 0$. These are infinite-dimensional analogs of theorems of P\'olya and Reznick. The proof relates evaluations of the (truncated) power sum map $(p_2,p_3,\dots)$ to moments of discrete probability measures on the compact interval $[-1,1]$. This yields a characterization of the closure of the orbit space of the infinite symmetric group on the sphere. Finally, we provide an alternative proof of existing Positivstellens\"atze for normalized symmetric functions.

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Sebastian Debus, Robin Schabert. 2026-06-25. Any-dimensional Positivstellens\"atze for symmetric functions. https://arxiv.org/abs/2606.27551

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