The Pythagoras number of a rational function field in two variables

We prove that every sum of squares in the rational function field in two variables \(K(X,Y)\) over a hereditarily pythagorean field \(K\) is a sum of \(8\) squares. More precisely, we show that the Pythagoras number of every finite extension of \(K(X)\) is at most \(5\). The main ingredients of the proof are a local-global principle for quadratic forms over function fields in one variable over a complete rank-\(1\) valued field due to V. Mehmeti and a valuation theoretic characterization of hereditarily pythagorean fields due to L. Br\"ocker.