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