On Hilbert's construction of positive polynomials

2007/07/14 by Bruce Reznick, Reznick, Bruce
Computer Science · Mathematics · #11E20 #11E25 #12D99 #14H50 #14N15 #Algebraic Geometry (math.AG) #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT) #Polynomial and algebraic computation #math.AG #math.NT #msc:11E20 #msc:11E25 #msc:12D99 #msc:14H50 #msc:14N15

paper · pdf · doi:10.48550/arxiv.0707.2156

arxiv created 2007/07/14 · openalex publication_date 2007/07/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 1888, Hilbert described how to find real polynomials in more than one variable which take only non-negative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert's construction and present many such polynomials.

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