Bounding nonminimality and a conjecture of Borovik–Cherlin

  • James Freitag

    University of Illinois at Chicago, USA
  • Rahim Moosa

    University of Waterloo, Canada
Bounding nonminimality and a conjecture of Borovik–Cherlin cover

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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 -rank plus 2. The Borovik–Cherlin conjecture itself is verified for algebraic and meromorphic group actions, and a bound of -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.

Cite this article

James Freitag, Rahim Moosa, Bounding nonminimality and a conjecture of Borovik–Cherlin. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1384