Hilbert's fourteenth problem and finite generation ideals

Hilbert's fourteenth problem asks whether invariant rings under algebraic group actions are always finitely generated. There are a number of examples that have been constructed since the mid-20th century which demonstrate that this is not the case in general. This thesis is concerned with developing our understanding of these non-finitely generated invariant rings. This goal is ambitious, as by their nature these rings are difficult to work with and it is hard to build an intuition for what might be true in general. The difficulty of trying to develop a solid intuition from examples is exacerbated by the process of ``removing symmetries,'' which relates some of the more well-understood invariant rings. A key construction we employ in order to better understand the structure of these counterexamples to Hilbert's problem is the finite generation ideal, consisting of invariants which make the invariant ring finitely generated after localisation.

We take a number of paths in order to achieve our aim, including computing the finite generation ideal for existing examples, constructing new counterexamples, and improving our understanding of both the process of removing symmetries and the finite generation ideal itself. Specifically, we first compute the finite generation ideal of a famous counterexample due to Daigle and Freudenburg. Next, we work on constructing new non-finitely generated invariant rings, focusing primarily on an example proposed by Maubach. We then investigate this process of removing symmetries on some new examples. Finally, we study the finite generation ideal in the setting of monomial algebras, with the intention of passing results obtained to SAGBI-bases; a form of generating set we employ to compute the finite generation ideal for invariant rings.