Symmetric nonnegative functions, the tropical Vandermonde cell and superdominance of power sums
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917744322019328 |
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| author | Acevedo, Jose Blekherman, Grigoriy Debus, Sebastian Riener, Cordian |
| author_facet | Acevedo, Jose Blekherman, Grigoriy Debus, Sebastian Riener, Cordian |
| contents | We study nonnegative and sums of squares symmetric (and even symmetric) functions of fixed degree. We can think of these as limit cones of symmetric nonnegative polynomials and symmetric sums of squares of fixed degree as the number of variables goes to infinity. We compare these cones, including finding explicit examples of nonnegative polynomials which are not sums of squares for any sufficiently large number of variables, and compute the tropicalizations of their dual cones in the even symmetric case. We find that the tropicalization of the dual cones is naturally understood in terms of the overlooked superdominance order on partitions. The power sum symmetric functions obey this same partial order (analogously to how term-normalized power sums obey the dominance order). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04616 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetric nonnegative functions, the tropical Vandermonde cell and superdominance of power sums Acevedo, Jose Blekherman, Grigoriy Debus, Sebastian Riener, Cordian Algebraic Geometry Combinatorics Optimization and Control 14P99, 05E05, 14T90, 05A17 We study nonnegative and sums of squares symmetric (and even symmetric) functions of fixed degree. We can think of these as limit cones of symmetric nonnegative polynomials and symmetric sums of squares of fixed degree as the number of variables goes to infinity. We compare these cones, including finding explicit examples of nonnegative polynomials which are not sums of squares for any sufficiently large number of variables, and compute the tropicalizations of their dual cones in the even symmetric case. We find that the tropicalization of the dual cones is naturally understood in terms of the overlooked superdominance order on partitions. The power sum symmetric functions obey this same partial order (analogously to how term-normalized power sums obey the dominance order). |
| title | Symmetric nonnegative functions, the tropical Vandermonde cell and superdominance of power sums |
| topic | Algebraic Geometry Combinatorics Optimization and Control 14P99, 05E05, 14T90, 05A17 |
| url | https://arxiv.org/abs/2408.04616 |