Hilbert's 17th problem

Hilbert's 17th problem asks whether every non-negative polynomial function—meaning it never produces negative values—can be expressed as a sum of squares of other polynomial functions. In simpler terms, it questions if such functions can always be broken down into combinations of squared terms. The problem was answered affirmatively in 1934 by mathematician Emil Artin, who proved that any non-negative polynomial can indeed be written as a sum of squares of rational functions. This result deepened understanding in algebra and real algebraic geometry, linking the structure of polynomials to their positivity properties.