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Sum-product theorems for polynomials

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If you have a question about this talk, please contact Boris Bukh.

Suppose A is a set of numbers and f(x,y) is a polynomial, how small can f(A,A) be? If f(x,y)=x+y or f(x,y)=xy, then f(A,A) can be very small indeed if A is a progression. However, Erdős and Szemerédi proved that A+A and AA cannot be simultaneously small when A is a set of real numbers. Their results has been generalized to other rings, and have found numerous applications in number theory, combinatorics, theoretical computer science, and other fields.

In this talk, I will survey the classical sum-product estimates, and will discuss several new results for other polynomial functions f. Joint work with Jacob Tsimerman.

This talk is part of the Discrete Analysis Seminar series.

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