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Motivated by the search for methods to establish strong minimality of certainlow order algebraic differential equations, a measure of how far a finite rankstationary type is from being minimal is introduced and studied: The {\emdegree of nonminimality} is the minimum number of realisations of the typerequired to witness a nonalgebraic forking extension. Conditional on the truthof a conjecture of Borovik and Cherlin on the generic multiple-transitivity ofhomogeneous spaces definable in the stable theory being considered, it is shownthat the nonminimality degree is bounded by the $U$-rank plus $2$. TheBorovik-Cherlin conjecture itself is verified for algebraic and meromorphicgroup actions, and a bound of $U$-rank plus $1$ is then deduced unconditionallyfor differentially closed fields and compact complex manifolds. An applicationis given regarding transcendence of solutions to algebraic differentialequations.

Bounding nonminimality and a conjecture of Borovik-Cherlin