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  "body": "\n## Problem\n\nConsider the equation of degree seven\n$$x^7 + a x^3 + b x^2 + c x + 1 = 0,$$\nand its solution $x$ as a function of the three coefficients $a, b, c$.\nCan this function be written as a finite [[composition|0THBK9V825]] of [[continuous functions|0X3WGX95FE]] of two variables?\n\n## Remarks\n\nHilbert expected the answer to be negative.\nEvery 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.\n\nIn 1957 Andrey Kolmogorov and Vladimir Arnold showed that the answer is positive for continuous functions, in a strong form.\nArnold proved that every continuous function of three variables is a composition of continuous functions of two variables.\nKolmogorov's superposition theorem goes further: every continuous $f \\colon [0, 1]^n \\to \\R$ can be written as\n$$f(x_1, \\dots, x_n) = \\sum_{q=0}^{2n} \\Phi_q\\Big( \\sum_{p=1}^{n} \\varphi_{q, p}(x_p) \\Big)$$\nwith continuous functions $\\Phi_q$ and $\\varphi_{q, p}$ of one variable.\nSo addition is the only function of two variables that is needed.\n\nMany authors read Hilbert's question as being about algebraic functions instead of continuous ones; in that form the problem is still open.\n\n## Explanations\n\n### L1 human\n\nSome functions of several variables secretly need only one or two variables at a time, such as $\\sin(x + y z)$, built from multiplication, addition and the sine.\nHilbert asked whether the solution of a particular seventh-degree equation, which depends on three numbers, can be built in this way from continuous functions of two variables.\nKolmogorov and Arnold showed that, surprisingly, every continuous function of any number of variables can be built from continuous functions of one variable and addition.\n",
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      "first_stated": "1900",
      "posed_by": "David Hilbert"
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    "name": "Hilbert's thirteenth problem",
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