有界非极小性与Borovik-Cherlin猜想
Bounding nonminimality and a conjecture of Borovik-Cherlin
摘要：
出于寻找建立某些低阶代数微分方程的强极小性的方法的动机，引入并研究了有限秩平稳类型离极小有多远的度量：{ \em非极小度}是见证非代数分叉扩张所需的类型的最小实现数。在Borovik和Cherlin关于稳定理论中定义的齐性空间的一般多重传递性的猜想成立的条件下，证明了非极小度是由$U$ -秩加$2$所限定的.证明了Borovik-Cherlin猜想本身在代数和亚纯群作用下的正确性，并无条件地导出了微分闭域和紧致复流形的$U$ -秩加$1$的界.给出了关于代数微分方程解的超越性的一个应用。
英文摘要：
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.
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