Bounding nonminimality and a conjecture of Borovik-Cherlin

TL;DR

This paper introduces a measure called the degree of nonminimality for stationary types in model theory, relates it to the U-rank, and verifies a conjecture for certain algebraic structures, with applications to differential equations.

Contribution

It defines the degree of nonminimality, establishes bounds under a conjecture, and verifies the conjecture for specific algebraic and geometric structures.

Findings

Nonminimality degree is bounded by U-rank plus 2 under the conjecture.

The Borovik-Cherlin conjecture is verified for algebraic and meromorphic group actions.

Unconditional bound of U-rank plus 1 is obtained for differentially closed fields and complex manifolds.

Abstract

Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The {\em degree of nonminimality} is the minimum number of realisations of the type required to witness a nonalgebraic forking extension. Conditional on the truth of a conjecture of Borovik and Cherlin on the generic multiple-transitivity of homogeneous spaces definable in the stable theory being considered, it is shown that the nonminimality degree is bounded by the $U$-rank plus $2$. The Borovik-Cherlin conjecture itself is verified for algebraic and meromorphic group actions, and a bound of $U$-rank plus $1$ is then deduced unconditionally for differentially closed fields and compact complex manifolds. An application is given regarding transcendence of solutions to algebraic differential equations.