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Preprint Number 1994

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1994. James Freitag and Rahim Moosa
Bounding nonminimality and a conjecture of Borovik-Cherlin

Submission date: 4 June 2021


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 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.

Mathematics Subject Classification: 03C45, 14L30, 12H05, 32J99

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Full text arXiv 2106.02537: pdf, ps.

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