Sums of squares and higgher powers in function fields

David Grimm
Universidad de Santiago de Chile
I will first give an overview over known results on upper and lower bounds on the minimal number of N of squares (or higher even powers)needed  in order to write every positive rational function on a real variety as a sum of N squares (or higher even powers if the function satisfies additional pole- and vanishing order criterias), and examplify how various techniques from algebraic and arithmetic geometry were used to obtain those bounds.
These results go back to Becker, Cassels, Ellison, Kucharz and Pfister. In the second part of the talk, I shall focus in particular on the result of Kucharz, which implies the lower bound 3 for function fields of surfaces over the real numbers, and I will present my own (more elementary) proof, which moreover extends the lower bound to situations where of non-real closed base fields.