Contributions to the model theory of algebraic differential equations

Abstract

This thesis deals with semiminimal analyses of finite rank types, primarily in the stable theory of differentially closed fields of characteristic zero (DCF0). The two main themes considered in this thesis are determining when a type is minimal or semiminimal, and understanding what invariants of finite rank types are captured by a semiminimal analysis.

In DCF0, a central concern of this thesis is determining when a type is almost internal to the field of constants. Partially generalising a result of Rosenlicht, algebraic criteria are provided in two different contexts: rational vector fields on affine n-space, and pullbacks under the logarithmic derivative of certain types which are internal to the constants. The criteria in the former case answers a question posed by Freitag, Jaoui, Marker and Nagloo about when the Poizat equations are internal to the constants. In both cases, the theory of binding groups in stable theories plays a significant role.

Results of Duan and Nagloo are improved upon to completely classify when the generic types of Lotka-Volterra systems are minimal. In the minimal case, a characterization of the possible relations that may exist between solutions of distinct Lotka-Volterra systems is given.

In the general setting of a totally transcendental theory, it is shown that the multiplicity with which a minimal type arises in a semiminimal analysis of a finite rank type is invariant, i.e., it is independent of the semiminimal analysis. A conjecture is proposed for the possible ways for two semiminimal analyses of the same finite rank type to differ. Along the way, the connection between semiminimal analyses and domination decompositions, is clarified.