Algebraic methods in sum-product phenomena

TL;DR

This paper classifies certain bivariate polynomials over the reals based on their sum-product behavior, establishing bounds that relate the size of sumsets and polynomial images of finite sets.

Contribution

It provides a classification of polynomials with sum-product type bounds and introduces new bounds linking sumsets and polynomial images over real numbers.

Findings

Established a lower bound: |A+A||f(A,A)|    |A|^{5/2}

Classified polynomials based on sum-product phenomena over  

Utilized Bezout's theorem and Stein's theorem in proofs

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.