Tropical Invariants for Permutation Group Actions

2025/12/15 by Harm Derksen, Derksen, Harm
Computer Science · Mathematics · #Polynomial and algebraic computation #Advanced Differential Equations and Dynamical Systems #Commutative Algebra and Its Applications

paper · pdf · doi:10.48550/arxiv.2512.13452

Abstract

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 Rⁿ/G into Euclidean space. We also show that the invariant polynomials of degree ≤ n p₁p₂⋯ p_k generate the semifield of invariant rational tropical functions, where p₁,p₂,…,p_k are the first k prime numbers. Most results are also true over arbitrary semirings that are additively idempotent and multiplicatively cancellative.

Citations

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Related

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• Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields
• Galois actions for semifield extensions and Galois coverings on tropical curves
• Idempotent Semigroups and Tropical Algebraic Sets
• Asymptotic computations of tropical refined invariants in genus 0 and 1