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