Algebraic methods in sum-product phenomena

Abstract

We classify the polynomials $f(x,y)\in \mathbb{R}[x,y]$ such that given any finite set $A\subset \mathbb{R}$ if $|A+A|$ is small, then $|f(A,A)|$ is large. In particular, the following bound holds : $|A+A||f(A,A)|\gtrsim |A{|}^{5/2}.$ The Bezout's theorem and a theorem by Y. Stein play important roles in our proof.

introduction revised