Hilbert's Fourteenth Problem

The fourteenth problem of Hilbert is the fourteenth of the problems posed by David Hilbert in his famous report at the II International Congress of Mathematicians in Paris in 1900. It is devoted to the issue of the final generation of rings arising under certain constructions. Hilbert’s original formulation was motivated by Maurer’s work, which stated the finite generation of the algebra of invariants of the linear action of an algebraic group on a vector space; in fact, Hilbert’s question concerned the ring obtained by the intersection of the subfield in the field of rational functions with the ring of polynomials. ^[one]

However, shortly after the report, it turned out that Maurer's work contained an error, and the Hilbert question began to be considered as a question of the finite generation of algebras of invariants of linear algebraic groups. Unexpectedly, the answer to this question turned out to be negative: in 1958, at the congress in Edinburgh, M. Nagata presented him with a counterexample ^[1] ^[2] . He constructed ^{[3] a} subgroup in GL (n) whose invariant algebra is not finitely generated. This construction was then simplified ^{[1] by} Steinberg in his 1997 paper ^[4] .

Content

1 Wording
- 1.1 The original formulation of Hilbert
- 1.2 Finite Generation of the Algebra of Invariants
2 Literature

Wording

Hilbert's original wording

14. Proof of finiteness of some complete system of functions.
<...> Maurer recently managed to extend the theorems of finiteness proved by Jordan and me in the theory of invariants to the case when the invariants are determined not by a general projective group, as in the ordinary theory of invariants, but by its arbitrary subgroup. <...>
Let some number m of entire rational functions be given $X_{1},...,X_{m}$ ${\ displaystyle X_ {1}, ..., X_ {m}}$ from variables $x_{1},\dots ,x_{n}$ ${\ displaystyle x_ {1}, \ dots, x_ {n}}$ :
$\left.{\begin{array}{c}X_{1}=f_{1}(x_{1},\dots ,x_{n})\\X_{2}=f_{2}(x_{1},\dots ,x_{n})\\\vdots \\X_{m}=f_{m}(x_{1},\dots ,x_{n})\end{array}}\right\}\qquad \qquad (S)$ ${\ displaystyle \ left. {\ begin {array} {c} X_ {1} = f_ {1} (x_ {1}, \ dots, x_ {n}) \\ X_ {2} = f_ {2} ( x_ {1}, \ dots, x_ {n}) \\\ vdots \\ X_ {m} = f_ {m} (x_ {1}, \ dots, x_ {n}) \ end {array}} \ right \} \ qquad \ qquad (S)}$
Every whole rational connection between $X_{1},\dots ,X_{m}$ ${\ displaystyle X_ {1}, \ dots, X_ {m}}$ if these their meanings are introduced into it, obviously, also represents an entire rational function of $x_{1},\dots ,x_{n}$ ${\ displaystyle x_ {1}, \ dots, x_ {n}}$ . However, there may well be fractional rational functions of $X_{1},\dots ,X_{m}$ ${\ displaystyle X_ {1}, \ dots, X_ {m}}$ which, after substituting (S), lead to entire functions of $x_{1},\dots ,x_{n}$ ${\ displaystyle x_ {1}, \ dots, x_ {n}}$ . Each such function <...> I will call a relatively entire function of $X_{1},\dots ,X_{m}$ ${\ displaystyle X_ {1}, \ dots, X_ {m}}$ . <...> The problem, therefore, is expressed in the following: to establish whether it is always possible to find such a finite system with respect to entire functions of $X_{1},\dots ,X_{m}$ ${\ displaystyle X_ {1}, \ dots, X_ {m}}$ through which any other relatively entire function is expressed in a whole and rational way. <...> ^[5]

In other words, it is a question of the finite generation of an algebra $K\bigcap k[x_{1},\dots ,x_{n}]$ ${\ displaystyle K \ bigcap k [x_ {1}, \ dots, x_ {n}]}$ where $K$ ${\ displaystyle K}$ - generated $X_{1},\dots ,X_{n}$ ${\ displaystyle X_ {1}, \ dots, X_ {n}}$ field. Since every intermediate field $k\subset K\subset k(x_{1},\dots ,x_{n})$ ${\ displaystyle k \ subset K \ subset k (x_ {1}, \ dots, x_ {n})}$ is finitely generated as an extension of k; as a result, in modern language, the original Hilbert formulation is as follows:

Let be $K\subset k[x_{1},\dots ,x_{n}]$ ${\ displaystyle K \ subset k [x_ {1}, \ dots, x_ {n}]}$ - some field containing the main field k. Is it true that algebra $K\bigcap k[x_{1},\dots ,x_{n}]$ ${\ displaystyle K \ bigcap k [x_ {1}, \ dots, x_ {n}]}$ of course originated? ^[one]

Finite Generation of the Algebra of Invariants

Literature

↑ ¹ ² ³ ⁴ Notes of the course of I. Arzhantsev “ Algebras of invariants and 14 Hilbert's problem ”
↑ Dieudonne J. , Carroll J., Mumford D. Geometric theory of invariants. - M., Mir, 1974. - c. 74-81
↑ M. Nagata, Lectures on the Fourteenth problem of Hilbert. Tata Institute, 1965.
↑ R. Steinberg, Nagata's example. In: "Algebraic Groups Lie Groups", Austral. Math. Soc. Lect. Series 9, Cambr. University Press (1997), 375-384.
↑ Translation of the Hilbert report from German - M. G. Shestopal and A. V. Dorofeev , published in the book Problems of Hilbert / ed. P.S. Aleksandrova . - M .: Nauka, 1969 .-- S. 45-47. - 240 p. - 10,700 copies. Archived on October 17, 2011. Archived October 17, 2011 on Wayback Machine

I.V. Arzhantsev. Graded algebras and the 14th Hilbert problem. - M .: ICMMO, 2009 .-- 64 p. - 1000 copies. - ISBN 978-5-94057-491-0 .
Hilbert Problems / ed. P.S. Aleksandrova . - M .: Nauka, 1969 .-- 240 p. - 10,700 copies. Archived on October 17, 2011. Archived October 17, 2011 on Wayback Machine

[arzh-1] ¹ ² ³ ⁴ Notes of the course of I. Arzhantsev “ Algebras of invariants and 14 Hilbert's problem ”

[2] Dieudonne J. , Carroll J., Mumford D. Geometric theory of invariants. - M., Mir, 1974. - c. 74-81

[3] M. Nagata, Lectures on the Fourteenth problem of Hilbert. Tata Institute, 1965.

[4] R. Steinberg, Nagata's example. In: "Algebraic Groups Lie Groups", Austral. Math. Soc. Lect. Series 9, Cambr. University Press (1997), 375-384.

[HilbertRus-5] Translation of the Hilbert report from German - M. G. Shestopal and A. V. Dorofeev , published in the book Problems of Hilbert / ed. P.S. Aleksandrova . - M .: Nauka, 1969 .-- S. 45-47. - 240 p. - 10,700 copies. Archived on October 17, 2011. Archived October 17, 2011 on Wayback Machine