M/E23

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 fields carrying a complete valuation of rank one due to Vlerë Mehmeti and a valuation theoretic characterization of hereditarily pythagorean fields due to Ludwig Bröcker. This is joint work with Nicolas Daans, David Grimm, Gonzalo Manzano Flores and Marco Zaninelli.

Karim Becher

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