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 DCF of differentially closed fields of characteristic zero with m distinguished commuting derivations. We show that if p is a minimal type relative to 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 DCFA of difference-differentially closed fields of characteristic zero with m commuting derivations in the sense that every completion has SU-rank one. We illustrate this theorem by describing in detail the possible difference-differential equations consistent with differential equations of the form y' = y3 + a2 y2 + a1 y + a0 where a0, a1, and a2 are differential constants.
Related
- Existence and Nonexistence of Solutions of Certain type of Nonlinear Differential and Differential-Difference Equations
- Explicit meromorphic solutions of certain types of nonlinear differential-difference equations
- Finite dimensional reductions of integrable differential-difference equations
- Entire solutions of Fermat-Type Partial Differential-Difference equations in higher dimensions
- Finite-Order Entire Solutions to Fermat-Type Partial Differential-Difference Systems in ℂn
BibTeX
@misc{li2026differential,
title = {{Differential equations as difference-differential equations}},
author = {Wei Li and Thomas Scanlon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22317},
eprint = {2610.22317},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22317},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Differential equations as difference-differential equations
AU - Wei Li
AU - Thomas Scanlon
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22317
UR - https://arxiv.org/abs/2610.22317
AB - 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 DCF of differentially closed fields of characteristic zero with m distinguished commuting derivations. We show that if p is a minimal type relative to 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 DCFA of difference-differentially closed fields of characteristic zero with m commuting derivations in the sense that every completion has 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.
ER -