2012/08/19 by Claus Scheiderer, Scheiderer, Claus · 1 citation
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Matrix Theory and Algorithms #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.1208.3865
openalex publication_date 2012/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the closed convex hull of any one-dimensional semi-algebraic subset of Rn has a semidefinite representation, meaning that it can be written as a linear projection of the solution set of some linear matrix inequality. This is proved by an application of the moment relaxation method. Given a nonsingular affine real algebraic curve C and a compact semialgebraic subset K of its R-points, the preordering P(K) of all regular functions on C that are nonnegative on K is known to be finitely generated. We prove that P(K) is stable, meaning that uniform degree bounds exist for weighted sum of squares representations of elements of P(K). We also extend this last result to the case where K is only virtually compact. The main technical tool for the proof of stability is the archimedean local-global principle. As a consequence of our results we prove that every convex semialgebraic subset of R2 has a semidefinite representation.