Closure of the Cone of Sums of 2d-powers in Certain Weighted ℓ1-seminorm Topologies
| Autor: | Murray Marshall, Sven Wagner, Mehdi Ghasemi |
|---|---|
| Rok vydání: | 2014 |
| Předmět: |
Pure mathematics
Representation theorem Semigroup General Mathematics Polynomial ring 010102 general mathematics Closure (topology) Primary 43A35 Secondary 44A60 13J25 Commutative Algebra (math.AC) Mathematics - Commutative Algebra 01 natural sciences Semigroup with involution Integer Cone (topology) 0103 physical sciences FOS: Mathematics 010307 mathematical physics 0101 mathematics Commutative property Mathematics |
| Zdroj: | Canadian Mathematical Bulletin. 57:289-302 |
| ISSN: | 1496-4287 0008-4395 |
| DOI: | 10.4153/cmb-2012-043-9 |
| Popis: | In a paper from 1976, Berg, Christensen, and Ressel prove that the closure of the cone of sums of squares in the polynomial ring in the topology induced by the ℓ1-norm is equal to Pos([–1; 1]n), the cone consisting of all polynomials that are non-negative on the hypercube [–1,1]n. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and 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 -seminorm topology associated with an absolute value. In this paper we give a new proof of these results, which is based on Jacobi’s representation theoremfrom2001. 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 in the topology induced by the -norm is equal to Pos([–1; 1]n). |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|