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  "body": "\n## Problem\n\nLet $n \\ge 1$ and let $f \\in \\R[x_1, \\dots, x_n]$ be a polynomial with real coefficients such that $f(a) \\ge 0$ for every $a \\in \\R^n$.\nIs $f$ a sum of squares of [[rational functions|0WCZH4AFMT]], that is, are there $g_1, \\dots, g_m \\in \\R(x_1, \\dots, x_n)$ with $f = g_1^2 + \\cdots + g_m^2$?\n\n## Remarks\n\nEmil Artin answered the question positively in 1927.\nHis proof uses the theory of [[formally real fields|X32TSWJSYT]] and [[real closures|778KFSM09P]].\nIf $f$ were not a sum of squares, some ordering of $\\R(x_1, \\dots, x_n)$ would make $f$ negative.\nPassing to a real closure then yields a real point where $f \u003c 0$.\nRational functions are needed: a nonnegative polynomial need not be a sum of squares of polynomials.\nAn example is the Motzkin polynomial $x^4 y^2 + x^2 y^4 - 3x^2 y^2 + 1$, which is not a sum of squares of polynomials.\nIt is nonnegative by the inequality of arithmetic and geometric means applied to $x^4 y^2$, $x^2 y^4$ and $1$.\nHilbert showed that every nonnegative polynomial is a sum of squares of polynomials exactly in these cases: one variable, degree $2$, or two variables and degree $4$.\nPfister showed that $2^n$ squares of rational functions always suffice.\n\n## Explanations\n\n### L1 human\n\nA sum of squares of real numbers is never negative.\nHilbert asked for the converse for polynomials: if a polynomial never takes a negative value, can one see this by writing it as a sum of squares?\nWith fractions allowed, the answer is yes, and a certificate of positivity always exists.\nWith polynomials only, the answer is no in general.\n",
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