Hilbert's fourteenth problem
Encyclopedia
In mathematics
, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems
proposed in 1900, asks whether certain ring
s 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 function
s in n variables,
Consider now the ring R defined as the intersection
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
such that every element in R can be rationally represented by the yi. But this does not imply that the ring R is finitely generated as a ring, even if all the elements yi 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
.
. 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[x1, ..., xn] (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 xi 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 number
s for some classical semi-simple Lie group
s (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.
'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 function
s on X is finitely generated over k.
Zariski's formulation was shown to be equivalent to the original problem, for X normal.
Several authors have reduced the sizes of the group and the vector space in Nagata's example. For example, showed that over any field there is an action of the sum G of 3 copies of the additive group on k18 whose ring of invariants is not finitely generated.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems
Hilbert's problems
Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...
proposed in 1900, asks whether certain ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
s are finitely generated
Finitely generated algebra
In mathematics, a finitely generated algebra is an associative algebra A over a field K where there exists a finite set of elements a1,…,an of A such that every element of A can be expressed as a polynomial in a1,…,an, with coefficients in K...
.
The setting is as follows: Assume that k is a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
and let K be a subfield of the field of rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s in n variables,
- k(x1, ..., xn ) over k.
Consider now the ring R defined as the intersection
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
- yi, i = 1 ,...,d in K
such that every element in R can be rationally represented by the yi. But this does not imply that the ring R is finitely generated as a ring, even if all the elements yi 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
Masayoshi Nagata
Masayoshi Nagata was a Japanese mathematician, known for his work in the field of commutative algebra....
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
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...
.
History
The problem originally arose in algebraic invariant theoryInvariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
. Here the ring R is given as a (suitably defined) ring of polynomial invariants of a linear algebraic group
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...
over a field k acting algebraically on a polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
k[x1, ..., xn] (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 xi 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
Arthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
, Sylvester
James Joseph Sylvester
James Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics...
, Clebsch
Alfred Clebsch
Rudolf Friedrich Alfred Clebsch was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. He subsequently taught in Berlin and Karlsruhe...
, Paul Gordan and also Hilbert) of invariants of binary forms in two variables with the natural action of the special linear group
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
SL2(k) on it. Hilbert himself proved the finite generation of invariant rings in the case of the field of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s for some classical semi-simple Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s (in particular the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
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
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
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
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
inside the polynomial ring generated by the invariants.
Zariski's formulation
ZariskiOscar Zariski
Oscar Zariski was a Russian mathematician and one of the most influential algebraic geometers of the 20th century.-Education:...
's formulation of Hilbert's fourteenth problem asks whether, for a quasi-affine
Quasiprojective variety
In mathematics, a quasiprojective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset...
algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
X over a field k, possibly assuming X normal
Normal scheme
In mathematics, in the field of algebraic geometry, a normal scheme is a scheme X for which every stalk of its structure sheaf OX is an integrally closed local ring; that is, each stalk is an integral domain such that its integral closure in its field of fractions is equal to itself.Any reduced...
or smooth, the ring of regular function
Regular function
In mathematics, a regular function is a function that is analytic and single-valued in a given region. In complex analysis, any complex regular function is known as a holomorphic function...
s on X is finitely generated over k.
Zariski's formulation was shown to be equivalent to the original problem, for X normal.
Nagata's counterexample
gave the following counterexample to Hilbert's problem. The field k is a field containing 48 elements a1i, ...,a16i, 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 the each of the three vectors (a1i, ...,a16i) 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 tj and taking xj to xj + bjtj. 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, showed that over any field there is an action of the sum G of 3 copies of the additive group on k18 whose ring of invariants is not finitely generated.