Hilbert's thirteenth problem
- Hilbert's thirteenth problem
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 7-th degree equations using 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. The actual question is more easily posed however in terms of continuous functions. Hilbert asked whether it was possible to construct the solution of the general seventh degree equation
:"x"7 + "ax"3 + "bx"2 + "cx" + 1 = 0
using a finite number of two-variable functions. A more general question is to ask: are there continuous functions of three variables which cannot be expressed as a composition of continuous functions of two variables?
The answer was given by Vladimir Arnold in 1957, 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 in fact only two-variable functions were required, thus answering Hilbert's question.
Arnold later returned to the question, jointly with Goro Shimura (V. I. Arnold and G. Shimura, "Superposition of algebraic functions" (1976), in "Mathematical Developments Arising From Hilbert's Problems").
References
*G. G. Lorentz, "Approximation of Functions" (1966), Ch. 11
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