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

EHilbert’s Seventh Problem: Express a nonnegative rational function as quotient of sums of squares.

Some polynomials with inputs in the real numbers always take non-negative values; an easy example is $x^2 + y^2$. Hilbert’s 17th problem asks whether such a polynomial can always be written as the sum of squares of rational functions (a rational function is the quotient of two polynomials). In 1927, Emil Artin solved the question in the affirmative.