Closure of the Cone of Sums of $2d$-powers in Certain Weighted $ell_1$-seminorm Topologies

Canadian mathematical bulletin

Sven Wagner¹  Mehdi Ghasemi²  Murray Marshall² 

[1] Technische Universität Dortmund, Fakultät für Mathematik, Lehrstuhl VI, Vogelpothsweg 87, 44227 Dortmund, Germany;Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK S7N 5E6

关键词: positive definite;    moments;    sums of squares;    involutive semigroups;   

DOI  :  10.4153/CMB-2012-043-9

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

In a paper from 1976, Berg, Christensen and Ressel prove that theclosure of the cone of sums of squares $summathbb{R}[underline{X}]^2$ in the polynomial ring$mathbb{R}[underline{X}] := mathbb{R}[X_1,dots,X_n]$ in thetopology induced by the $ell_1$-norm is equal to$operatorname{Pos}([-1,1]^n)$, the cone consisting of all polynomialswhich are non-negative on the hypercube $[-1,1]^n$. The result isdeduced as a corollary of a general result, established in the samepaper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen and Resselestablish an even more general result, for a commutative semigroupwith involution, for the closure of the cone of sums of squares ofsymmetric elements in the weighted $ell_1$-seminorm topologyassociated to an absolute value. In the present paper we give a new proof of these results which isbased on Jacobi's representation theorem from 2001. At the same time,we use Jacobi's representation theorem to extend these results fromsums of squares to sums of $2d$-powers, proving, in particular, thatfor any integer $dge 1$, the closure of the cone of sums of$2d$-powers $sum mathbb{R}[underline{X}]^{2d}$ in$mathbb{R}[underline{X}]$ in the topology induced by the$ell_1$-norm is equal to $operatorname{Pos}([-1,1]^n)$.

【 授权许可】

Unknown