Model Theory of Complex Numbers with Polynomial Functions

Abstract

Let $\mathbb C$ be the set of complex numbers, and let $\mathcal P$ be a collection of complex polynomial maps in several variables. Assuming at least one $P\in\mathcal P$ depends on at least two variables, we classify all possibilities for the structure $(\mathscr M;\mathcal P)$ up to definable equivalence. In particular, outside a short list of exceptions, we show that $(\mathscr M;\mathcal P)$ always defines $+$ and $\times$. Our tools include Zilber's Restricted Trichotomy, as well as the classification of symmetric non-expanding pairs of polynomials over $\mathbb C$ from arithmetic combinatorics. Along the way, we also give a new condition for a reduct $\mathscr M=(M,...)$ of a smooth curve over an algebraically closed field to recover all constructible subsets of powers of $M$.