Recent progress on Hilbert's Fourteenth Problem via triangular derivations (Q2724122)

A special case of Hilbert's fourteenth problem asks whether the kernel of a \(k\)-derivation \(D\) on the polynomial ring \(k[x_1,x_2,\dots,x_n]\) is always finitely generated (where \(k\) a field of characteristic 0). By results of Zariski, this is true for \(n\leq 3\). The reviewer showed that Nagata's famous counterexample to Hilbert's fourteenth problem can be used to construct a derivation on the polynomial ring in 32 variables whose kernel is not finitely generated. Smaller and easier counterexamples have been found by Roberts (\(n=7\)), Freudenburg (\(n=6\)) and by Daigle and Freudenburg (\(n=5\)). The only case open is \(n=4\). All the new counterexamples have a triangular form (and are locally nilpotent). Daigle and Freudenburg showed that the kernel of a triangular derivation in four variables is finitely generated. This paper gives an overview about the finite generation of kernels of derivations with triangular form.