Carathéodory numbers for measures supported on genus one curves

We investigate the problem of representing a Borel measure supported on an elliptic normal curve, when restricted to bounded degree polynomials, as the sum of Dirac measures. The smallest number of Dirac measures needed to represent any such measure is called the Carathéodory number. This number governs the complexity of cubature rules and can be interpreted as the rank of Waring-type minimal representations with nonnegative coefficients.

Despite its importance, and several asymptotic results, no exact values for the Carathéodory numbers were known beyond the rational case. In this talk, we show how in the genus one case this number depends on the topology of the real locus of the supporting curve, exploiting the duality with nonnegative polynomials.

Based on a joint work with Greg Blekherman and Rainer Sinn.