Hilbert's fourteenth problem and field modifications

TL;DR

This paper investigates Hilbert's fourteenth problem, demonstrating the existence of new counterexamples in the case where the extension degree is 2 and the number of variables is 3, based on the properties of intermediate fields.

Contribution

It introduces a field-theoretic approach to construct new counterexamples to Hilbert's fourteenth problem, especially for the case with extension degree 2 and three variables.

Findings

Existence of counterexamples with extension degree 2 and three variables.

Minimality of intermediate fields leads to counterexamples.

Automorphisms can produce minimal counterexamples.

Abstract

Let $k({\bf x})=k(x_1,\ldots ,x_n)$ be the rational function field, and $k\subsetneqq L\subsetneqq k({\bf x})$ an intermediate field. Then, Hilbert's fourteenth problem asks whether the $k$-algebra $A:=L\cap k[x_1,\ldots ,x_n]$ is finitely generated. Various counterexamples to this problem were already given, but the case $[k({\bf x}):L]=2$ was open when $n=3$. In this paper, we study the problem in terms of the field-theoretic properties of $L$. We say that $L$ is minimal if the transcendence degree $r$ of $L$ over $k$ is equal to that of $A$. We show that, if $r\ge 2$ and $L$ is minimal, then there exists $\sigma \in {\mathop{\rm Aut}\nolimits}_kk(x_1,\ldots ,x_{n+1})$ for which $\sigma (L(x_{n+1}))$ is minimal and a counterexample to the problem. Our result implies the existence of interesting new counterexamples including one with $n=3$ and $[k({\bf x}):L]=2$.