Rank and Dimension in Difference-Differential Fields
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.
Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1320427645
Digital Object Identifier: doi:10.1215/00294527-1499363
Zentralblatt MATH identifier: 05991007
Mathematical Reviews number (MathSciNet): MR2855879