Submission date: 3 August 2023

Abstract:

Let ℂ be the set of complex numbers, and let PP be a collection of complex polynomial maps in several variables. Assuming at least one P ∈ PP depends on at least two variables, we classify all possibilities for the structure (M ; PP) up to definable equivalence. In particular, outside a short list of exceptions, we show that (M ; PP) 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.

Mathematics Subject Classification: Primary 0C345, Secondary 11B30

Keywords and phrases:

Full text arXiv 2308.01632: pdf, ps.