The degree of nonminimality is at most two

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.