Model theory of complex numbers with polynomial functions

• Benjamin Castle ^[2] ; Chieu-Minh Tran ^[1]
  1. [1] National University of Singapore

     National University of Singapore

     Singapur

     
  2. [2] Department of Mathematics, University of Illinois Urbana Champaign, Champaign, IL, United States
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 5, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01086-x
• Enlaces
  • Texto completo
• Resumen

  • 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.

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