On Hilbert's construction of positive polynomials

By Bruce Reznick. 2007

63 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/c2b59fa46f1a35f47a6e819caeecc6cc771444bf

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In Computational Theory and Mathematics

In Geometry and Topology

In Theoretical Computer Science

Citations in the corpus

Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems (2009, methodology): “[16, 17]) shows that (Hn,d) holds if and only if n = 1, n = d = 2, or n > 1 and d = 1.”
Algebraic boundaries of Hilbert’s SOS cones (2011, background): “t concerning ∂P3,6\Σ3,6. All exposed extreme rays are sextics with ten acnodes, and all extreme rays are limits of sextics with ten acnodes. This proves the second part of Reznick’s Conjecture 7.9 in [23]. Indeed, in the second paragraph of the above proof we saw that C has ten ordinary singularitie”
Nonnegative Polynomials and Sums of Squares (2010, background): “Let P ′ and Sq′ be the images of Pn,2d and Σn,2d under E, respectively, and let H ′ be the image of Hn,2d.”