### Rank and Dimension in Difference-Differential Fields

Ronald F. Bustamante Medina
Source: Notre Dame J. Formal Logic Volume 52, Number 4 (2011), 403-414.

#### Abstract

Hrushovski proved that the theory of difference-differential fields of characteristic zero has a model-companion, which we shall denote DCFA. Previously, the author proved that this theory is supersimple. In supersimple theories there is a notion of rank defined in analogy with Lascar U-rank for superstable theories. It is also possible to define a notion of dimension for types in DCFA based on transcendence degree of realization of the types. In this paper we compute the rank of a model of DCFA, give some properties regarding rank and dimension, and give an example of a definable set with finite rank but infinite dimension. Finally we prove that for the case of definable subgroup of the additive group being finite-dimensional and having finite rank are equivalent.

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Primary Subjects: 11U09, 12H05, 12H10