Moments, Sums of Squares, and Tropicalization
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arXiv
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| Format: | Preprint |
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2022
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| author | Blekherman, Grigoriy Rincón, Felipe Sinn, Rainer Vinzant, Cynthia Yu, Josephine |
| author_facet | Blekherman, Grigoriy Rincón, Felipe Sinn, Rainer Vinzant, Cynthia Yu, Josephine |
| contents | We use tropicalization to study the duals to cones of nonnegative polynomials and sums of squares on a semialgebraic set $S$. The truncated cones of moments of measures supported on the set $S$ is dual to nonnegative polynomials on $S$, while "pseudo-moments" are dual to sums of squares approximations to nonnegative polynomials. We provide explicit combinatorial descriptions of tropicalizations of the moment and pseudo-moment cones, and demonstrate their usefulness in distinguishing between nonnegative polynomials and sums of squares. We give examples that show new limitations of sums of squares approximations of nonnegative polynomials. When the semialgebraic set is defined by binomial inequalites, its moment and pseudo-moment cones are closed under Hadamard product. In this case, their tropicalizations are polyhedral cones that encode all binomial inequalities on the moment and pseudo-moment cones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_06291 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Moments, Sums of Squares, and Tropicalization Blekherman, Grigoriy Rincón, Felipe Sinn, Rainer Vinzant, Cynthia Yu, Josephine Algebraic Geometry Combinatorics Functional Analysis Optimization and Control We use tropicalization to study the duals to cones of nonnegative polynomials and sums of squares on a semialgebraic set $S$. The truncated cones of moments of measures supported on the set $S$ is dual to nonnegative polynomials on $S$, while "pseudo-moments" are dual to sums of squares approximations to nonnegative polynomials. We provide explicit combinatorial descriptions of tropicalizations of the moment and pseudo-moment cones, and demonstrate their usefulness in distinguishing between nonnegative polynomials and sums of squares. We give examples that show new limitations of sums of squares approximations of nonnegative polynomials. When the semialgebraic set is defined by binomial inequalites, its moment and pseudo-moment cones are closed under Hadamard product. In this case, their tropicalizations are polyhedral cones that encode all binomial inequalities on the moment and pseudo-moment cones. |
| title | Moments, Sums of Squares, and Tropicalization |
| topic | Algebraic Geometry Combinatorics Functional Analysis Optimization and Control |
| url | https://arxiv.org/abs/2203.06291 |