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Pré-Publication, Document De Travail Année : 2024

Nonnegative Polynomials and Moment Problems on Algebraic Curves

Résumé

The cone of nonnegative polynomials is of fundamental importance in real algebraic geometry, but its facial structure is understood in very few cases. We initiate a systematic study of the facial structure of the cone of nonnegative polynomials $P$ on a smooth real projective curve $X$. We show that there is a duality between its faces and totally real effective divisors on $X$. This allows us to fully describe the face lattice in case $X$ has genus 1. The dual cone $P^\vee$ is known as the moment cone, and it plays an important role in real analysis. We compute the Carathéodory number of $P^\vee$ for an elliptic normal curve $X$, which measures the complexity of quadrature rules of measures supported on $X$. This number can also be interpreted as a maximal typical Waring rank with nonnegative coefficients. Interestingly, the topology of the real locus of $X$ influences the Carathéoodory number of $P^\vee$. We apply our results to truncated moment problems on affine cubic curves, where we deduce sharp bounds on the flat extension degree.
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Dates et versions

hal-04639107 , version 1 (08-07-2024)
hal-04639107 , version 2 (09-07-2024)

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  • HAL Id : hal-04639107 , version 2

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Lorenzo Baldi, Grigoriy Blekherman, Rainer Sinn. Nonnegative Polynomials and Moment Problems on Algebraic Curves. 2024. ⟨hal-04639107v2⟩
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