Hilbert's fourteenth problem and field modifications

Abstract

Let $k(\mathbf{x})=k(x_1,\dots ,x_n)$ be the rational function field, and $k⫋L⫋k(\mathbf{x})$ an intermediate field. Then, Hilbert's fourteenth problem asks whether the $k$-algebra $A:=L\cap k[x_1,\dots ,x_n]$ is finitely generated. Various counterexamples to this problem were already given, but the case $[k(\mathbf{x}):L]=2$ 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 $\sigma \in {\mathrm{Aut}}_{k}k(x_1,\dots ,x_{n+1})$ 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 $[k(\mathbf{x}):L]=2$.