# Zero-Separating Invariants for Linear Algebraic Groups

Jonathan Elmer, Martin Kohls

Research output: Contribution to journalArticle

1 Citation (Scopus)

## Abstract

Let G be a linear algebraic group over an algebraically closed field acting rationally on a G-module V with its null-cone. Let δ(G, V) and σ(G, V) denote the minimal number d such that for every and, respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) ≠ 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V. For positive characteristics, we show that δ(G) = â for any subgroup G of GL2() that contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G is unipotent. Our results also lead to a more elementary proof that β sep(G) is finite if and only if G is finite.

Original language English 911-924 14 Proceedings of the Edinburgh Mathematical Society 59 4 22 Dec 2015 https://doi.org/10.1017/S0013091515000322 Published - 1 Nov 2016

## Keywords

• geometrically reductive
• global degree bounds
• invariant theory
• linear algebraic groups
• prime characteristic