  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.
