Nonnegative Polynomials and Moment Problems on Algebraic Curves

TL;DR

This paper explores the structure of nonnegative polynomials on algebraic curves, revealing dualities and computing complexity measures, with applications to moment problems on cubic curves.

Contribution

It systematically characterizes the facial structure of nonnegative polynomial cones on algebraic curves and computes the Carathéodory number for elliptic curves.

Findings

Complete description of face lattice for genus one curves

Carathéodory number computed for elliptic normal curves

Bounds on flat extension degree for moment problems on cubic curves

Abstract

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 $\pos$ 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 one. We compute the Carath\'{e}odory number of the dual moment cone $\pos^\vee$ for an elliptic normal curve $X$, which measures the complexity of quadrature rules of measures supported on $X$. Interestingly, the topology of the real locus of $X$ influences the Carath\'{e}odory number of $\pos^\vee$. We apply our results to truncated moment problems on affine cubic curves, where we deduce sharp bounds on the flat extension degree.