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arXiv · 2603.10455

Positivity of polynomials on the nonnegative part of certain affine hypersurfaces

Colin Tan·Wing-Keung To
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Abstract

We consider polynomials on the intersection of the closed positive orthant with the height-$1$ level hypersurface of certain polynomials with positive coefficients. We show that any polynomial strictly positive on such a semi-algebraic set can be represented by some polynomial with only positive coefficients. This result generalizes a result of P\'olya which corresponds to the case when the semi-algebraic set is the standard simplex. Our proof uses the Archimedean Representation Theorem from real algebra.

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Colin Tan, Wing-Keung To. 2026-03-11. Positivity of polynomials on the nonnegative part of certain affine hypersurfaces. https://arxiv.org/abs/2603.10455

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