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A Survey of Counterexamples to Hilbert's Fourteenth Problem

Freudenburg, Gene — 2001

Serdica Mathematical Journal

We survey 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. We also include 8 important open questions.

Recent progress on Hilbert's Fourteenth Problem via triangular derivations

Gene Freudenburg — 2001

Annales Polonici Mathematici

We give an overview of recent results concerning kernels of triangular derivations of polynomial rings. In particular, we examine the question of finite generation in dimensions 4, 5, 6, and 7.

A nonlinearizable action of $S_{3}$ on $ℂ^{4}$

Gene Freudenburg; Lucy Moser-Jauslin — 2002

Annales de l’institut Fourier

The main purpose of this article is to give an explicit algebraic action of the group $S_{3}$ of permutations of 3 elements on affine four-dimensional complex space which is not conjugate to a linear action.

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