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MODEL THEORY OF DIFFERENCE FIELDS, II: PERIODIC IDEALS AND THE TRICHOTOMY IN ALL CHARACTERISTICS

Published online by Cambridge University Press:  02 August 2002

ZOÉ CHATZIDAKIS ,
EHUD HRUSHOVSKI  and
YA'ACOV PETERZIL
ZOÉ CHATZIDAKIS
Affiliation:
UFR de Mathématiques, Université Paris 7, Case 7012, 2, place Jussieu, 75251 Paris Cedex 05, France. zoe@logique.jussieu.fr
EHUD HRUSHOVSKI
Affiliation:
Institute of Mathematics, The Hebrew University, Givat Ram, 91904 Jerusalem, Israel. ehud@math.huji.ac.il
YA'ACOV PETERZIL
Affiliation:
Department of Mathematics, University of Haifa, 31905 Haifa, Israel. kobi@mathcs2.haifa.ac.il
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Abstract

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. We show 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)

Keywords

model theory applied to algebradifference fields
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 85 , Issue 2 , September 2002 , pp. 257 - 311
Copyright
2002 London Mathematical Society

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