Model theory of complex numbers with polynomial functions Model theory of complex numbers with polynomial functions

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...

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Vydané v:Selecta mathematica (Basel, Switzerland) Ročník 31; číslo 5
Hlavní autori: Castle, Benjamin, Tran, Chieu-Minh
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Cham Springer International Publishing 01.11.2025
Springer Nature B.V
Predmet:
Combinatorial analysis
Complex numbers
Fields (mathematics)
Functions (mathematics)
Mathematics
Mathematics and Statistics
Polynomials
Secondary 11B30
Primary 03C45
ISSN:1022-1824, 1420-9020
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Popis
Shrnutí: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 of a smooth curve over an algebraically closed field to recover all constructible subsets of powers of M .
Bibliografia:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1022-1824
1420-9020
DOI:10.1007/s00029-025-01086-x