Seminar

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A central tool in additive combinatorics is the "polynomial method," a family of powerful techniques for studying the existence and size of sets satisfying given properties by encoding them in terms of the zeros of certain polynomials, which can then be analyzed from an algebraic perspective.



In this talk, we introduce some iconic classical and modern versions of the polynomial method—including the Combinatorial Nullstellensatz, the Croot-Lev-Pach method, and Dvir's method of multiplicities—from the unified perspective of a new framework based on what we call "shift operators." We show how to take advantage of the many useful properties of these operators to rederive the core results of these existing versions of the polynomial method. We also touch on some novel directions in which these new tools may be fruitfully applied.