Kolloquien und Oberseminare Übersicht Detail

Vortragende Person/Vortragende Personen:
Karim Becher

The Pythagoras number of a commutative ring K is the smallest natural number n such that every sum of squares of elements of K is a sum of n squares of elements of K, or infinity, if no such number n exists.  The study of this ring invariant, denoted by p(K), is a very classical topic of algebra, and it has in particular been a crucial incentive for the development of the theory of quadratic forms over fields. Nevertheless, very basic questions on Pythagoras numbers of fields remain unsolved until today. For example, very little is known about the growth of the Pythagoras number under field extensions.

In my talk, I want to present this problem and some known results. This will include the following result obtained in a joint work by Nicolas Daans, David Grimm, Gustavo Manzano-Flores, Marco Zaninelli and myself. In the late 1970s, Eberhard Becker had characterised the fields K for which p(K(X))=2 holds. Based on an intriguing local-global principle for quadratic forms over function fields due to Vlerë Mehmeti, we have now been able to show that the condition that p(K(X))=2 implies the upper bound p(F)\leq 5 for function fields in one variable F/K, and furthermore the bound p(K(X,Y))\leq 8.