Nonnegative Polynomials and Moment Problems on Algebraic Curves

Overview & Synopsis

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.