Model Theory of Complex Numbers with Polynomial Functions

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 $(\mathcal{M};\mathcal{P})$ up to definable equivalence. In particular, outside a short list of exceptions, we show that $(\mathcal{M};\mathcal{P})$ 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 $\mathcal{M}=(M,\mathrm{.}\mathrm{.}\mathrm{.})$ of a smooth curve over an algebraically closed field to recover all constructible subsets of powers of $M$.