Model theory of difference fields, II: Periodic ideals and the trichotomy in all characteristics

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)