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

摘要

The model theory of difference fields was investigated in [4]. A dimension theory was described, including a decomposition into one-dimensional definable sets. While the general theory was developed in all characteristics, the deepest result was only obtained in characteristic 0. The result in question, Zil'ber's Trichotomy, is a classification of the possible combinatorial geometries underlying the one-dimensional sets; we refer the reader to the introduction to [4] or to the body of this paper for a discussion. This paper complements [4] by giving a completely new proof of the trichotomy, valid in all characteristics.