Resumen de Model theory of complex numbers with polynomial functions

Benjamin Castle, Chieu Minh Tran

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