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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 or not 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

\( x^7 + ax^3 + bx^2 + cx + 1 = 0 \)

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...")(see, for example, Abhyankar: Shreeram S. Abhyankar: Hilbert's Thirteenth Problem, Vitushkin: A. G. Vitushkin: On Hilbert's thirteenth problem and related questions, Chebotarev: (N. G. Chebotarev, "On certain questions of the problem of resolvents")). 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 (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).