A note on the representation of positive polynomials with structured sparsity

By David Grimm, Tim Netzer, Markus Schweighofer. 2006

47 citations recorded by the source corpus.

Abstract.We consider real polynomials in finitely many variables. Let the variables consist of finitely many blocks that are allowed to overlap in a certain way. Let the solution set of a finite system of polynomial inequalities be given, where each inequality involves only variables of one block. We investigate polynomials that are positive on such a set and sparse in the sense that each monomial involves only variables of one block. In particular, we derive a short and direct proof for Lasserre’s theorem on the existence of sums of squares certificates respecting the block structure. The motivation for the results can be found in the literature on numerical methods for global optimization of polynomials that exploit sparsity.

Subfields: Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9cba7dacacfb240e4cf5e05365d115f92157df02

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In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Real algebraic geometry (2013, methodology): “[18] Let γ : V → R be a parametrized path such that U = π1.”
Sums of Squares, Moment Matrices and Optimization Over Polynomials (2009, methodology): “The following proof of Theorem 5.14 is from Schweighofer [134]; it is less technical than the proof just presented, but uses some basic knowledge about Gröbner bases.”