Harm Derksen, University of Michigan
Algorithms for Invariant Rings

The main goal in Invariant Theory is to study the ring of all polynomials that are invariant under an group action. Invariant rings are not always finitely generated, because of Nagata's counterexample to Hilbert fourteenth problem. By results of Hilbert, Nagata and Haboush, invariant rings of reductive groups are finitely generated. Unfortunately, most finite generation proofs are not constructive. In particular, they do not provide algorithms for finding a set of generators for the invariant ring. In this talk I will discuss various algorithms for generators of invariant rings. I will also present recent results of Gregor Kemper and myself: We found the first algorithm for generators of invariant rings of reductive groups actions on affine varieties in arbitrary characteristic. We also found an algorithm for generators of invariant rings for unipotent group actions on the polynomial ring. In that case, the ring of invariants may not be finitely generated, but the output of the algorithm presents the ring of invariants as the ring of regular functions on some explicitly given quasi-affine variety.

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