Tropical Invariants for Permutation Group Actions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917147152744448 |
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| author | Derksen, Harm |
| author_facet | Derksen, Harm |
| contents | We consider the action of a permutation group $G$ of order $k$ on the tropical polynomial semiring in $n$ variables. We prove that the sub-semiring of invariant polynomials is finitely generated if and only if $G$ is generated by $2$-cycles. There do exist finitely many separating invariants of degree at most $\max\{n,{n\choose 2}\}$. Separating tropical invariants can be used to construct bi-Lipschitz embeddings of the orbit space ${\mathbb R}^n/G$ into Euclidean space. We also show that the invariant polynomials of degree $\leq n p_1p_2\cdots p_k$ generate the semifield of invariant rational tropical functions, where $p_1,p_2,\dots,p_k$ are the first $k$ prime numbers. Most results are also true over arbitrary semirings that are additively idempotent and multiplicatively cancellative. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_13452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tropical Invariants for Permutation Group Actions Derksen, Harm Commutative Algebra Algebraic Geometry Combinatorics 13A50, 14T10 (Primary) 12K10, 15A80 (Secondary) We consider the action of a permutation group $G$ of order $k$ on the tropical polynomial semiring in $n$ variables. We prove that the sub-semiring of invariant polynomials is finitely generated if and only if $G$ is generated by $2$-cycles. There do exist finitely many separating invariants of degree at most $\max\{n,{n\choose 2}\}$. Separating tropical invariants can be used to construct bi-Lipschitz embeddings of the orbit space ${\mathbb R}^n/G$ into Euclidean space. We also show that the invariant polynomials of degree $\leq n p_1p_2\cdots p_k$ generate the semifield of invariant rational tropical functions, where $p_1,p_2,\dots,p_k$ are the first $k$ prime numbers. Most results are also true over arbitrary semirings that are additively idempotent and multiplicatively cancellative. |
| title | Tropical Invariants for Permutation Group Actions |
| topic | Commutative Algebra Algebraic Geometry Combinatorics 13A50, 14T10 (Primary) 12K10, 15A80 (Secondary) |
| url | https://arxiv.org/abs/2512.13452 |