2024/07/08 by Lorenzo Baldi, Baldi, Lorenzo, Grigoriy Blekherman +3 · 4 citations
Computer Science · Engineering · Mathematics · #14H52 #14P25 #14P99 #44A60 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2407.06017
openalex publication_date 2024/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Carathéodory 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 Carathéodory 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.