### 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 |
---|---|

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 |

### Fingerprint

### 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

**Zero-Separating Invariants for Linear Algebraic Groups.** / Elmer, Jonathan; Kohls, Martin.

Research output: Contribution to journal › Article

*Proceedings of the Edinburgh Mathematical Society*, vol. 59, no. 4, pp. 911-924. https://doi.org/10.1017/S0013091515000322

}

TY - JOUR

T1 - Zero-Separating Invariants for Linear Algebraic Groups

AU - Elmer, Jonathan

AU - Kohls, Martin

N1 - This paper was prepared during visits of M.K. to the University of Aberdeen and of J.E. to T. U. M¨unchen. The first of these visits was supported by the Edinburgh Mathematical Society’s Research Support Fund. We want to thank Gregor Kemper and the Edinburgh Mathematical Society for making these visits possible. We also thank the anonymous referee for useful suggestions.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - 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.

AB - 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.

KW - geometrically reductive

KW - global degree bounds

KW - invariant theory

KW - linear algebraic groups

KW - prime characteristic

UR - http://www.scopus.com/inward/record.url?scp=84951283527&partnerID=8YFLogxK

U2 - 10.1017/S0013091515000322

DO - 10.1017/S0013091515000322

M3 - Article

AN - SCOPUS:84951283527

VL - 59

SP - 911

EP - 924

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 4

ER -