Hilbert's fourteenth problem

In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain rings are finitely generated.

The setting is as follows: Assume that "k" is a field and let "K" be a subfield of the field of rational functions in "n" variables,

:"k"("x"₁, ..., "x"_"n" ) over "k".

Consider now the ring "R" defined as the intersection

:$R:=\; K\; cap\; k\; [x\_1,\; dots,\; x\_n]\; .$

Hilbert conjectured that all such subrings are finitely generated. It can be shown that the field "K" is always finitely generated "as a field", in other words, there exist finitely many elements

:"y"_"i", "i" = 1 ,...,"d" in "K"

such that every element in "R" can be "rationally" represented by the "y"_"i". But this does not imply that the ring "R" is finitely generated "as a ring", even if all the elements "y"_i could be chosen from "R".

After some results were obtained confirming Hilbert's conjecture in special cases and for certain classes of rings (in particular the conjecture was proved unconditionally for "n" = 1 and "n" = 2 by Zariski in 1954) then in 1959 Masayoshi Nagata found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants for the action of a linear algebraic group.

History

The problem originally arose in algebraic invariant theory. Here the ring "R" is given as a (suitably defined) ring of polynomial invariants of a linear algebraic group over a field "k" acting algebraically on a polynomial ring "k" ["x"₁, ..., "x"_n] (or more generally, on a finitely generated algebra defined over a field). In this situation the field "K" is the field of "rational" functions (quotients of polynomials) in the variables "x"_i which are invariant under the given action of the algebraic group, the ring "R" is the ring of "polynomials" which are invariant under the action. A classical example in 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 with the natural action of the special linear group "SL₂(k)" on it. Hilbert himself proved the finite generation of invariant rings in the case of the field of complex numbers for some classical semi-simple Lie groups (in particular the general linear group over the complex numbers) and specific linear actions on polynomial rings, i.e. actions coming from finite-dimensional representations of the Lie-group. This finiteness result was later extended by Hermann Weyl to the class of all semi-simple Lie-groups. 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's formulation of Hilbert's fourteenth problem asks whether, for a quasi-affine algebraic variety "X" over a field "k", possibly assuming "X" normal or smooth, the ring of regular functions on "X" is finitely generated over "k".

Zariski's formulation was shown [cite journal | last = Winkelmann | first = Jörg | title = Invariant rings and quasiaffine quotients | journal = Math. Z. | volume = 244 | issue = 1 | pages = 163–174 | date = 2003 | doi = 10.1007/s00209-002-0484-9] to be equivalent to the original problem, for "X" normal.

References

* M. Nagata, "On the Fourteenth Problem of Hilbert", Proceedings of the International Congress of Mathematicians 1958, pp. 459-462, Cambridge University Press.
* M. Nagata: "Lectures on the fourteenth problem of Hilbert". Lect. Notes 31, Tata Inst. Bombay, 1965.
* O. Zariski, "Interpretations algebrico-geometriques du quatorzieme probleme de Hilbert", Bulletin des Sciences Mathematiques 78 (1954), pp. 155-168.