Hilbert's 17th Problem: Expression of Definite Forms

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\ge1$, let $\mathbb R(x_1,\ldots,x_n)$ be the field of rational functions in $n$ independent variables with real coefficients. A rational function $f=p/q$, where $p,q\in\mathbb R[x_1,\ldots,x_n]$ and $q\ne0$, is nonnegative if $p(a)/q(a)\ge0$ for every $a\in\mathbb R^n$ with $q(a)\ne0$. This condition is independent of the chosen representation. A sum of squares of rational functions means a finite sum whose equality to $f$ holds in the rational-function field.

Statement recorded in the supplied source.

For every $n\ge1$ and every nonnegative $f\in\mathbb R(x_1,\ldots,x_n)$, do there exist $m\ge1$ and $r_1,\ldots,r_m\in\mathbb R(x_1,\ldots,x_n)$ such that \[ f=\sum_{j=1}^{m}r_j^2\quad\text{in }\mathbb R(x_1,\ldots,x_n)? \] [S1][S2] The polynomial formulation asks the same question for a polynomial nonnegative on all of $\mathbb R^n$. It is equivalent: if $f=p/q$ is nonnegative on $q\ne0$, then the polynomial $pq$ is nonnegative there, hence everywhere by continuity, and a rational-square representation of $pq$ can be divided by $q^2$. Artin answered this historical question affirmatively in 1927. [S2]

Short English statement

Can every real rational function that is nonnegative wherever defined be written as a finite sum of squares of real rational functions? Artin's solution of Hilbert's seventeenth problem gives an affirmative answer.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 90 (HIL-017). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Valéry Mahé, Using hyperelliptic curves to find positive polynomials that are not a sum of three squares in $\mathbb R(x,y)$, arXiv:math/0703722v2, 13 September 2007, introduction, p. 2: definition of nonnegative polynomials, the rational-square formulation, and Artin's 1927 affirmative answer. https://arxiv.org/pdf/math/0703722v2.