The List of Hilbert’s Twenty-Three Problems: Hilbert’s Problem #13

Seventh-Degree Polynomials: Solve 7th degree equation using algebraic (variant: continuous) functions of two parameters.

Hilbert’s 13th problem is about equations of the form 

$x^7 + ax^3 + bx^2 + cx + 1 = 0$. 

He asked whether solutions to these functions can be written as the composition of finitely many two-variable functions. (Hilbert believed they could not be.) In 1957, Andrey Kolmogorov and Vladimir Arnold proved that each continuous function of n variables — including the case in which n = 7 — can be written as a composition of continuous functions of two variables. However, if stricter conditions than mere continuity are imposed on the functions, the question is still open.

Πηγή: abakcus

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