Hilbert's thirteenth problem

Remarks

Hilbert expected the answer to be negative. Every equation of degree seven can be brought to the form above by Tschirnhaus transformations, so the question asks whether the general septic is solvable by functions of two variables.

In 1957 Andrey Kolmogorov and Vladimir Arnold showed that the answer is positive for continuous functions, in a strong form. Arnold proved that every continuous function of three variables is a composition of continuous functions of two variables. Kolmogorov's superposition theorem goes further: every continuous $f \colon [0, 1]^n \to \R$ can be written as $$f(x_1, \dots, x_n) = \sum_{q=0}^{2n} \Phi_q\Big( \sum_{p=1}^{n} \varphi_{q, p}(x_p) \Big)$$ with continuous functions $\Phi_q$ and $\varphi_{q, p}$ of one variable. So addition is the only function of two variables that is needed.

Many authors read Hilbert's question as being about algebraic functions instead of continuous ones; in that form the problem is still open.