Density of sums of three squares

Hilbert has proven that a real polynomial in two variables that takes only nonnegative values is a sum of four squares of rational functions. Cassels, Ellison and Pfister have shown that this result is optimal: there exist such polynomials that are not sums of three squares of rational functions. In this talk, we will explain why those polynomials that can be written as sums of three squares are dense in the set of those that are nonnegative. The proof relies on the study of real Noether–Lefschetz loci.