Sections of high rank varieties and applications

I will describe some recent 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 application is the following theorem : Let $V$ be a complex vector space, $P$ a high rank polynomial of degree $d$, and $X$ the null set of $P$, $X=\{v \mid P(v)=0\}$. Any function $f:X\to C$ which is polynomial of degree $d$ on lines in $X$ is the restriction of a degree $d$ polynomial on $V$.