Premium
Model theory of difference fields, II: Periodic ideals and the trichotomy in all characteristics
Author(s) -
Chatzidakis Zoé,
Hrushovski Ehud,
Peterzil Ya'acov
Publication year - 2002
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611502013576
Subject(s) - trichotomy (philosophy) , reduct , mathematics , limit (mathematics) , automorphism , pure mathematics , discrete mathematics , combinatorics , mathematical analysis , rough set , data mining , philosophy , linguistics , computer science
We classify all possible combinatorial geometries associated with one‐dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. Weshow that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one‐dimensional sets. This limit structure is shown to be a Zariski geometry in (roughly) the sense of Hrushovski and Zil'ber. The trichotomy is thus obtained for the limit structure as a consequence of a general theorem, and then shown to be inherited by the original theory. 2000 Mathematical Subject Classification : 03C60; (primary) 03C45, 03C98, 08A35, 12H10 (secondary)