Abstract
Invariant theory is concerned with the ring of polynomial functions that are left unchanged under the linear coordinate changes given by a group of invertible matrices. Hilbert showed in 1890 that this invariant ring is finitely generated for many matrix groups, including finite groups and polynomial representations of the special linear group. On the other hand, in 1959 Nagata exhibited group actions where this Noetherian property does not hold. In this tutorial we discuss these classical examples, with emphasis on their geometric meaning, and we present computer algebra tools for computing invariants.