Finite-dimensional differential-algebraic permutation groups

TL;DR

This paper characterizes finite rank definably primitive permutation groups in differentially closed fields, showing they are algebraic groups in the constants, and applies these results to conjectures and transcendence problems in differential algebra.

Contribution

It proves that finite rank definably primitive permutation groups are algebraic groups in the constants, advancing the understanding of their structure in differential algebraic geometry.

Findings

Finite rank definably primitive permutation groups are algebraic in the constants.

Verification of finite Morley rank permutation group conjectures in differentially closed fields.

Nonorthogonality implies non-weak orthogonality of higher Morley powers of types.

Abstract

Several structural results about permutation groups of finite rank definable in differentially closed fields of characteristic zero (and other similar theories) are obtained. In particular, it is shown that every finite rank definably primitive permutation group is definably isomorphic to an algebraic permutation group living in the constants. Applications include the verification, in differentially closed fields, of the finite Morley rank permutation group conjectures of Borovik-Deloro and Borovik-Cherlin. Applying the results to binding groups for internality to the constants, it is deduced that if complete types $p$ and $q$ are of rank $m$ and $n$, respectively, and are nonorthogonal, then the $(m+3)$rd Morley power of $p$ is not weakly orthogonal to the $(n+3)$rd Morley power of $q$. An application to transcendence of generic solutions of pairs of algebraic differential equations is given.