Closure of the cone of sums of 2d-powers in certain weighted $\ell_1$-seminorm topologies

TL;DR

This paper proves that in certain weighted $ ext{l}_1$-seminorm topologies, the closure of the cone of sums of 2d-powers equals the cone of polynomials non-negative on the hypercube, extending previous results on sums of squares.

Contribution

It provides a new proof using Jacobi's theorem and extends the closure characterization from sums of squares to sums of 2d-powers in weighted $ ext{l}_1$-seminorm topologies.

Findings

Closure of sums of 2d-powers cone equals non-negative polynomials on hypercube

New proof based on Jacobi's representation theorem

Extension of previous sums of squares results to sums of 2d-powers

Abstract

Berg, Christensen and Ressel prove that the closure of the cone of sums of squares in the ring of real polynomials in the topology induced by the $\ell_1$-norm is equal to the cone consisting of all polynomials which are non-negative on the hypercube $[-1,1]^n$. The result is deduced as a corollary of a general result which is valid for any commutative semigroup. In later work Berg and Maserick and also Berg, Christensen and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted $\ell_1$-seminorm topology associated to an absolute value. In the present paper we give a new proof of these results which is based on Jacobi's representation theorem. At the same time, we use Jacobi's representation theorem to extend these results from sums of squares to sums of 2d-powers, proving, in particular, that for any integer $d>0$, the closure of the cone of sums of 2d-powers in the ring of real polynomials in the topology induced by the $\ell_1$-norm is equal the cone consisting of all polynomials which are non-negative on the hypercube $[-1,1]^n$.