Model Theory of Complex Numbers with Polynomial Functions

TL;DR

This paper classifies the structures formed by complex numbers with polynomial functions, showing that most such structures define addition and multiplication, using tools from model theory and arithmetic combinatorics.

Contribution

It provides a comprehensive classification of structures involving complex polynomials, identifying when they define fundamental operations and introducing new conditions for reconstructing constructible sets.

Findings

Most polynomial-augmented structures define + and ×

A classification of symmetric non-expanding polynomial pairs over ℂ

A new criterion for reconstructing all constructible subsets from reducts

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 $M$.