Elsevier

Journal of Complexity

Volume 21, Issue 6, December 2005, Pages 823-844
Journal of Complexity

Non-existence of degree bounds for weighted sums of squares representations

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Abstract

Given a fixed family of polynomials h1,,hrR[x1,,xn], we study the problem of representing polynomials in the form(*)f=s0+s1h1++srhrwith sums of squares si. Let M be the cone of all f which admit such a representation. The problem is said to be stable if there exists a function ϕ:NN such that every fM has a representation (*) with deg(si)ϕ(deg(f)). The main result says that if the subset K={h10,,hr0} of Rn has dimension 2 and the sequence h1,,hr has the moment property (MP), then the problem is not stable. In particular, this includes the case where K is compact, dim(K)2 and the cone M is multiplicatively closed.

Keywords

Non-negative polynomials
Sums of squares
Complexity
Moment problem
Real algebraic geometry

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1

Support by DFG travel Grant KON 1823/2002 and by the European RAAG network HPRN-CT-2001-00271 is gratefully acknowledged. Part of this work was done while the author enjoyed a stay at MSRI Berkeley. He would like to thank the institute for the invitation and the very pleasant working conditions.