CLOSURE OF THE CONE OF SUMS OF 2d-POWERS IN CERTAIN WEIGHTED `1-SEMINORM TOPOLOGIES

In [3] Berg, Christensen and Ressel prove that the closure of the cone of sums of squares ∑R[X]2 in the polynomial ring R[X] := R[X1, . . . , Xn] in the topology induced by the `1-norm is equal to Pos([−1, 1]n), 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, also established in [3], which is valid for any commutative semigroup. In later work Berg and Maserick [5] and Berg, Christensen and Ressel [4] 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 `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 [13]. 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 ≥ 1, the closure of the cone of sums of 2d-powers ∑R[X]2d in R[X] in the topology induced by the `1-norm is equal to Pos([−1, 1]n).