Model theory of complex numbers with polynomial functions

Abstract Let $$\mathbb {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> be the set of complex numbers, and let $$\mathcal P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> be a collection of complex polynomial maps in several variables. Assuming at least one $$P\in \mathcal P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> depends on at least two variables, we classify all possibilities for the structure $$(\mathbb {C};\mathcal P)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>;</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> up to definable equivalence. In particular, outside a short list of exceptions, we show that $$(\mathbb {C};\mathcal P)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>;</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> always defines $$+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>+</mml:mo> </mml:math> and $$\times $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>×</mml:mo> </mml:math> . Our tools include Zilber’s Restricted Trichotomy, as well as the classification of symmetric non-expanding pairs of polynomials over $$\mathbb C$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> from arithmetic combinatorics. Along the way, we also give a new condition for a reduct "Equation missing" of a smooth curve over an algebraically closed field to recover all constructible subsets of powers of M .