A survey of counterexamples to Hilbert's fourteenth problem (Q2761689)

Hilbert's fourteenth problem is the following: For a field \(k\), let \(k^{[n]}\) denote the polynomial ring in \(n\) variables over \(k\) and let \(k^{(n)}\) denote its field of fractions. If \(K\) is a subfield of \(k^{(n)}\) containing \(k\), is \(K\cap k^{[n]}\) finitely generated over \(k\)? The problem is motivated by a fundamental question of invariant theory: If \(G\subset GL_n({\mathbb C})\) is a group of linear transformations and \(R\) is the polynomial ring in \(n\) variables over the complex field \(\mathbb C\), is the subring \(R^G\) of \(G\)-invariant polynomials finitely generated as a \(\mathbb C\)-algebra? The paper under review surveys some of the most remarkable classical and recent counterexamples to Hilbert's fourteenth problem, beginning with those of Nagata in the late 1950s, and including recent counterexamples in low dimension constructed with locally nilpotent derivations. Historical framework and pertinent references are provided. The author also includes some important open questions. NEWLINENEWLINENEWLINEThe paper is very interesting and very well-written in an attractive style. It will be useful for algebraists and acceptable for a large class of mathematicians.