Differential equations as difference-differential equations

Abstract

We analyze the behavior of systems of algebraic differential equations when considered as systems of difference-differential equations, with special emphasis on systems which define minimal sets relative to the theory $\operatorname{DCF}$ of differentially closed fields of characteristic zero with $m$ distinguished commuting derivations. We show that if $p$ is a minimal type relative to $\operatorname{DCF}$ defined by a system of algebraic partial differential equations and inequations and the forking geometry on $p$ is trivial (also called ``disintegrated''), then $p$ remains minimal when regarded as a partial type relative to the theory $\operatorname{DCFA}$ of difference-differentially closed fields of characteristic zero with $m$ commuting derivations in the sense that every completion has $\operatorname{SU}$-rank one. We illustrate this theorem by describing in detail the possible difference-differential equations consistent with differential equations of the form $y' = y^3 + a_2 y^2 + a_1 y + a_0$ where $a_0$, $a_1$, and $a_2$ are differential constants.