2009/11/06 by Victoria Powers, Powers, Victoria · 2 citations
Computer Science · Mathematics · #11E25 #13J30 #14P10 #14Q20 #Advanced Differential Equations and Dynamical Systems #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.0911.1331
openalex publication_date 2009/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Schmüdgen's Theorem says that if a basic closed semialgebraic set K = g1 ≥ 0, ..., gs ≥ 0 in Rn is compact, then any polynomial f which is strictly positive on K is in the preordering generated by the gi's. Putinar's Theorem says that under a condition stronger than compactness, any f which is strictly positive on K is in the quadratic module generated by the gi's. In this note we show that if the gi's and the f have rational coefficients, then there is a representation of f in the preordering with sums of squares of polynomials over Q. We show that the same is true for Putinar's Theorem as long as we include among the generators a polynomial N - ∑ Xi2, N a natural number.