The finite generation ideal for Daigle & Freudenberg's counterexample to Hilbert's fourteenth problem

TL;DR

This paper computes the finite generation ideal for a specific counterexample to Hilbert's fourteenth problem, revealing the structure of the invariant ring and its deviation from finite generation.

Contribution

It introduces a method to compute the finite generation ideal using SAGBI-bases, applied to Daigle and Freudenburg's counterexample.

Findings

Finite generation ideal is the radical of an ideal generated by three invariant families.

These three families plus an additional invariant form a SAGBI-basis.

The SAGBI-basis properties facilitate the computation of the finite generation ideal.

Abstract

We compute the finite generation ideal for Daigle and Freudenburg's counterexample to Hilbert's fourteenth problem. This ideal helps to understand how far the ring of invariants is from being finitely generated. Our calculations show that the finite generation ideal is the radical of an ideal generated by three infinite families of invariants. We show that these three families together with an additional invariant form a SAGBI-basis. We use the properties of our SAGBI-basis in our computation of the finite generation ideal.