Wei Li : Differential Equations as Difference-differential Equations

Posted by Ronnie Nagloo , part of the Logic Seminar.

At

Nov. 4, 2025, 3 p.m.

In

636 SEO

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 strongly minimal sets relative to the theory $DCF_{0,n}$ of differentially closed fields of characteristic zero with $n$ distinguished commuting derivations. We show that if $X$ is a strongly minimal set relative to $DCF_{0,n}$ defined by a finite system of algebraic partial differential equations and the forking geometry on $X$ is geometrically trivial, then $X$ remains minimal when regarded as definable set relative to the theory $DCFA_{0,n}$ of difference-differentially closed fields of characteristic zero with $n$ commuting derivations. We illustrate this theorem by describing in detail the possible difference-differential equations consistent with differential equations of the form $y' = f(y)$ for cubic polynomial $f$ over constants.