The degree of nonminimality is at most two

TL;DR

This paper investigates the nonminimality degree of types in differentially closed fields, showing that types of Lascar rank at least two have nonalgebraic forking extensions involving at most two realizations, with specific results when constants are involved.

Contribution

It establishes an upper bound of two on the degree of nonminimality for certain types in differentially closed fields, extending the understanding of forking extensions.

Findings

Types of Lascar rank at least two have nonalgebraic forking extensions involving at most two realizations.

If the type's parameters are in the constants, only one realization suffices for a nonalgebraic forking extension.

Results are applicable in a broader model-theoretic setting.

Abstract

It is shown that if $p$ is a complete type of Lascar rank at least 2 over $A$, in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations, $a_1$ and $a_2$, such that $p$ has a nonalgebraic forking extension over $A,a_1,a_2$. Moreover, if $A$ is contained in the field of constants then $p$ already has a nonalgebraic forking extension over $A,a_1$. The results are also formulated in a more general setting.