In mathematics, Hilbert's fourteenth problem, the fourteenth of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated.

be a field be a subfield of the field of rational functions

-algebra

.

for the transcendence degree, . Masayoshi Nagata constructed the first counterexamples in 1958, one of which he announced at the 1958 International Congress of Mathematicians.

in 2005..

History

The problem originally arose in algebraic invariant theory is given as a (suitably defined) ring of polynomial invariants of a linear algebraic group acting algebraically on a polynomial ring (or more generally, on a finitely generated algebra defined over a field).

is the ring of polynomials which are invariant under the action.

A classical example in the nineteenth century was the extensive study (in particular by Cayley, Sylvester, Clebsch, Paul Gordan and also Hilbert) of invariants of binary forms in two variables under the natural action of the special linear group.

Hilbert proved finite-generation results for invariant rings arising from important classical group actions in characteristic zero. These results were subsequently extended, notably by Hermann Weyl, and in modern form invariant rings of reductive algebraic groups acting on finitely generated algebras are finitely generated. By contrast, invariant rings of non-reductive groups need not be finitely generated; Nagata's counterexample is a suitably constructed ring of invariants for a linear action of a non-reductive linear algebraic group.

A major ingredient in Hilbert's proof is the Hilbert basis theorem applied to the ideal inside the polynomial ring generated by the invariants.

Zariski's formulation

Zariski gave an algebro-geometric formulation of Hilbert's fourteenth problem in terms of quasi-affine varieties. More precisely, the rings of global regular functions. Thus Hilbert's question can equivalently be viewed as asking whether the ring of global regular functions on such a quasi-affine variety must be finitely generated.

. See also Zariski's finiteness theorem.

Nagata's counterexample

Nagata constructed the first counterexamples to Hilbert's fourteenth problem in 1958. The following example was announced at the 1958 International Congress of Mathematicians.

and let

. Let

orthogonal to each of

by

.

.

Later counterexamples

Further counterexamples were constructed with progressively fewer variables. . by Gene Freudenburg by Daniel Daigle and Freudenburg.

is sharp...

In the linear invariant-theoretic setting, whose invariant rings are not finitely generated. with a non-finitely generated invariant ring.

See also

• Locally nilpotent derivation

• Nagata's conjecture on curves

References

1. Bibliography

   Footnotes