Hilbert's fourteenth problem

In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras 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(x₁, ..., x_n ) over k.

Consider now the k-algebra R defined as the intersection

${\displaystyle R:=K\cap k[x_{1},\dots ,x_{n}]\ .}$

Hilbert conjectured that all such algebras are finitely generated over k.

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's formulation was shown[1] to be equivalent to the original problem, for X normal. (See also: Zariski's finiteness theorem.)

Éfendiev F.F. (Fuad Efendi) provided symmetric algorithm generating basis of invariants of n-ary forms of degree r.^[2]

Nagata's counterexample

Nagata (1960) gave the following counterexample to Hilbert's problem. The field k is a field containing 48 elements a_1i, ...,a_16i, for i=1, 2, 3 that are algebraically independent over the prime field. The ring R is the polynomial ring k[x₁,...,x₁₆, t₁,...,t₁₆] in 32 variables. The vector space V is a 13-dimensional vector space over k consisting of all vectors (b₁,...,b₁₆) in k¹⁶ orthogonal to each of the three vectors (a_1i, ...,a_16i) for i=1, 2, 3. The vector space V is a 13-dimensional commutative unipotent algebraic group under addition, and its elements act on R by fixing all elements t_j and taking x_j to x_j + b_jt_j. Then the ring of elements of R invariant under the action of the group V is not a finitely generated k-algebra.

Several authors have reduced the sizes of the group and the vector space in Nagata's example. For example,

See also

• Locally nilpotent derivation

References

Bibliography

• MR 0116056, archived from the original
  on 2011-07-17
• MR 0215828
• Totaro, Burt (2008), "Hilbert's 14th problem over finite fields and a conjecture on the cone of curves", Compositio Mathematica, 144 (5): 1176–1198,
  MR 2457523
• O. Zariski, Interpretations algebrico-geometriques du quatorzieme probleme de Hilbert, Bulletin des Sciences Mathematiques 78 (1954), pp. 155–168.