“Hilbert's thirteenth problem”的意思、由来-开放百科全书

词条

Hilbert's thirteenth problem

释义

Introduction
History
See also
References

Hilbert's thirteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails proving whether a solution exists for all 7th-degree equations using algebraic (variant: continuous) functions of two arguments. It was first presented in the context of nomography, and in particular "nomographic construction" — a process whereby a function of several variables is constructed using functions of two variables. Hilbert's conjecture, that it is not always possible to find such a solution, was disproven in 1957.

Introduction

Hilbert considered the seventh-degree equation

and asked whether its solution, x, considered as a function of the three variables a, b and c, can be expressed as the composition of a finite number of two-variable functions.

History

Hilbert originally posed his problem for algebraic functions (Hilbert 1927, "...Existenz von algebraischen Funktionen...", i.e., "...existence of algebraic functions..."; also see Abhyankar 1997, Vitushkin 2004). However, Hilbert also asked in a later version of this problem whether there is a solution in the class of continuous functions.

A generalization of the second ("continuous") variant of the problem is the following question: can every continuous function of three variables be expressed as a composition of finitely many continuous functions of two variables? The affirmative answer to this general question was given in 1957 by Vladimir Arnold, then only nineteen years old and a student of Andrey Kolmogorov. Kolmogorov had shown in the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering the Hilbert's question when posed for the class of continuous functions.

Arnold later returned to the algebraic version of the problem, jointly with Goro Shimura (Arnold and Shimura 1976).

References

Shreeram S. Abhyankar, "Hilbert's Thirteenth Problem", Algèbre non commutative, groupes quantiques et invariants (Reims, 1995), 1–11, Sémin. Congr., 2, Soc. Math. France, Paris, 1997.
V. I. Arnold and G. Shimura, Superposition of algebraic functions (1976), in Mathematical Developments Arising From Hilbert Problems, Volume 1, Proceedings of Symposia in Pure Mathematics 28 (1976), pp. 45-46.
D. Hilbert, "Über die Gleichung neunten Grades", Math. Ann. 97 (1927), 243–250
G. G. Lorentz, Approximation of Functions (1966), Ch. 11
A. G. Vitushkin, "On Hilbert's thirteenth problem and related questions", Uspekhi Mat. Nauk 59:1 (2004), 11 24. (Translation in Russian Math. Surveys 59 (2004), no. 1, 11–25 )

3 : Polynomials|Hilbert's problems|Disproved conjectures

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

Introduction

History

See also

References