Sums of squares and higgher powers in function fields |
David Grimm |
Universidad de Santiago de Chile |
Abstract: |
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. |