A new shift operator-based polynomial method in additive combinatorics

We introduce a new form of the polynomial method based on what we call "shift operators," which we use to give efficient and intuitive new proofs of results previously shown using a wide range of polynomial methods, including Alon's Combinatorial Nullstellensatz and the Croot-Lev-Pach method. We end by discussing some potential new directions in which the tools introduced here may be fruitfully applied.