High strength varieties and applications

Tamar Ziegler, Hebrew University and IAS
Fine Hall 322

I will describe work with D. Kazhdan where we obtain results in algebraic geometry, inspired by questions in additive combinatorics, via analysis over finite fields. Specifically we are interested in quantitative properties of polynomial rings that are independent of the number of variables. A sample problem is the following : Let V be a complex vector space, P a polynomial of degree d, and X the null set of P,  X={v|P(v)=0}. Consider a function f:X —> C which is polynomial of degree d on all lines in X. When is f the restriction of a degree d polynomial on V? A key tool is a universality property satisfied by high strength varieties.