# A Survey of Counterexamples to Hilbert's Fourteenth Problem

@article{Freudenburg2001ASO, title={A Survey of Counterexamples to Hilbert's Fourteenth Problem}, author={Gene Freudenburg}, journal={Serdica. Mathematical Journal}, year={2001}, volume={27}, pages={171-192} }

We survey counterexamples to Hilbert's Fourteenth Problem, beginning with those of Nagata in the late 1950s, and including recent coun- terexamples in low dimension constructed with locally nilpotent derivations. Historical framework and pertinent references are provided. We also include 8 important open questions.

## 17 Citations

On the construction of quotient spaces by algebraic foliations

- Mathematics
- 2021

Given a variety defined over a field of characteristic zero and an algebraically integrable foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on…

The fourteenth problem of Hilbert for polynomial derivations

- Mathematics
- 2002

We present some facts, observations and remarks concerning the problem of finiteness of the rings of constants for derivations of polynomial rings over a commutative ring k containing the field Q of…

Computing with matrix invariants

- Mathematics
- 2005

This is an improved version of the talk of the author given at the Antalya Algebra Days VII on May 21, 2005. We present an introduction to the theory of the invariants under the action of GL(n,C) by…

THE NOETHER NUMBER IN INVARIANT THEORY

- Mathematics
- 2006

Let F be any field. Let G be any reductive linear algebraic group and consider a finite dimensional rational representation V of G. Then the Falgebra F[V ] of polynomial invariants for G acting on V…

Constructing quotients of algebraic varieties by linear algebraic group actions

- Mathematics
- 2015

In this article we review the question of constructing geometric quotients of actions of linear algebraic groups on irreducible varieties over algebraically closed fields of characteristic zero, in…

ACTA MATHEMATICA VIETNAMICA

- Mathematics
- 2011

Let f : R → R be a polynomial of degree d with only isolated complex critical points. It is shown that the set of bifurcation values of f is contained in a set which has at most (d − 1) points. The…

On Affine Embeddings of Reductive Groups on Affine Embeddings of Reductive Groups Acknowledgments

- Mathematics
- 2011

In this thesis, we study the properties and the classification of embeddings of homogeneous spaces, especially the case of affine normal embeddings of reductive groups. We might guess that as in the…

Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras

- Mathematics
- 2005

Affine embeddings of homogeneous spaces

- Mathematics
- 2000

Let G be a reductive algebraic group and let H be a reductive subgroup of G. We describe all pairs (G,H) such that for any affine G-variety X with a dense G-orbit isomorphic to G/H the number of…

## References

SHOWING 1-10 OF 45 REFERENCES

A Counterexample to Hilbert's Fourteenth Problem in Dimension 5☆

- Mathematics
- 1999

Abstract The kernel of a certain triangular derivation of the polynomial ring k [ x 1 , x 2 , x 3 , x 4 , x 5 ] is shown to be non-finitely generated over k (a field of characteristic zero), thus…

On some properties of elementary derivations in dimension six

- Mathematics
- 2001

An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert's fourteenth problem

- Mathematics
- 1990

On Weitzenböck’s theorem in positive characteristic

- Mathematics
- 1977

Let k be an algebraically closed field and letf: Ga GL(V) be a finite-dimensional k-rational representation of the additive group Ga. If the subspace of Ga-fixed points in V is a hyperplane, then the…

Computation of Invariants for Reductive Groups

- Mathematics, Computer Science
- 1999

An algorithm for computing generators of the invariant ring for a given representation of a linearly reductive group and a connection between some open conjectures in commutative algebra and finding good degree bounds for generating invariants is shown.

Polynomial Automorphisms and the Jacobian Conjecture

- Mathematics
- 2000

In this paper we give an update survey of the most important results concerning the Jacobian conjecture: several equivalent descriptions are given and various related conjectures are discussed. At…

Triangular Derivations of k[X1, X2, X3, X4]

- Mathematics
- 2001

Abstract It is shown that if k is a field of characteristic zero, then the kernel of any triangular k -derivation of k [ X 1 , X 2 , X 3 , X 4 ] is finitely generated as a k -algebra. This is…