### 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 |
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Pages (from-to) | 911-924 |

Number of pages | 14 |

Journal | Proceedings of the Edinburgh Mathematical Society |

Volume | 59 |

Issue number | 4 |

Early online date | 22 Dec 2015 |

DOIs | |

Publication status | Published - 1 Nov 2016 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Proceedings of the Edinburgh Mathematical Society*,

*59*(4), 911-924. https://doi.org/10.1017/S0013091515000322