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arXiv · 2604.18876

Geometry of numbers and degree bounds for rational invariants

Ben Blum-Smith·Sylvan Crane·Karla Guzman·Alexis Menenses·Maxine Song-Hurewitz
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Abstract

We investigate degree bounds for fields of rational invariants of representations of finite groups. We prove many cases of a bound for $\mathbb{Z}/p\mathbb{Z}$ conjectured by Blum-Smith, Garcia, Hidalgo, and Rodriguez. For arbitrary groups, we also prove a new bound on the minimum degree $d$ such that the polynomials of degree $\leq d$ span the field of rational functions as a vector space over the invariant field. This latter quantity also bounds the degree $d$ such that the polynomials of degree $\leq d$ contain a copy of the regular representation of $G$, advancing an inquiry of Kollár and Tiep. The methods involve Euclidean lattices and Minkowski's geometry of numbers.

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Ben Blum-Smith, Sylvan Crane, Karla Guzman, Alexis Menenses, Maxine Song-Hurewitz. 2026-04-20. Geometry of numbers and degree bounds for rational invariants. https://arxiv.org/abs/2604.18876

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