Abstract:  Let K be a complete discretely valued field with residue field k.  I will discuss the following theorem in my talk in Talca.  Suppose there is a constant A such that for each r, every system of r quadratic forms defined over k in more than Ar variables has a nontrivial zero over k.  Then every system of r quadratic forms defined over K in more than 2Ar variables has a nontrivial zero over K.   Using the well known results of Tsen-Lang, this theorem allows the computation of the u-invariant of an arbitrary function field defined over K.  

The proof of this theorem is based on four ideas.  One of them is well known (Hensel's Lemma), two are not as well known (Greenberg's theorem and the reduction process of Birch and Lewis), and the fourth idea is new.

In this talk I will discuss these ideas and give some of the background that is needed to prove the theorem above.