M/E23

The Pythagoras number of a field is the smallest number p such that every positive definite element in the field is a sum of p squares. It is an open problem whether the Pythagoras number p(K(X)) of a rational function field K(X) can be bounded in terms of p(K), the Pythagoras number of the base field. For the case when K is a number field, Pourchet proved in 1971 that p(K(X)) is at most 5. This bound is also sharp for example if K is the field of rational numbers Q, because X^2+7 is a sum of 5 but not of 4 squares in Q(X). Based on a detailed investigation of Pourchet's proof by my PhD student M. Zaninelli I will give an overview on the ingredients of this proof and point out some possible extensions.

Karim Becher

Antwerpen