Computing invariants of algebraic group actions in arbitrary characteristic

TL;DR

This paper introduces algorithms for computing invariants of algebraic group actions on affine varieties, covering reductive, unipotent, and non-finitely generated cases, with techniques for handling complex algebraic structures.

Contribution

It provides new algorithms for invariant ring computation in arbitrary characteristic, including non-finitely generated cases and methods for factorial varieties.

Findings

Algorithms for reductive group invariants

Methods for unipotent group invariants

Techniques for non-finitely generated algebras

Abstract

Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring K[X]^G in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes K[X]^G in terms of a so-called colon-operation. From this, generators of K[X]^G can be obtained in finite time if it is finitely generated. Under the additional hypothesis that K[X] is factorial, we present an algorithm that finds a quasi-affine variety whose coordinate ring is K[X]^G. Along the way, we develop some techniques for dealing with non-finitely generated algebras. In particular, we introduce the finite generation locus ideal.