Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and of some of its partial derivatives in $u$. DDEs occur frequently in combinatorics, especially in map enumeration. If a DDE is of fixed-point type then its solution $F(t,u)$ is unique, and a general result by Popescu (1986) implies that $F(t,u)$ is an algebraic power series. Constructive proofs of algebraicity for solutions of fixed-point type DDEs were proposed by Bousquet-M\'elou and Jehanne (2006). Bostan et. al (2022) initiated a systematic algorithmic study of such DDEs of order 1. We generalize this study to DDEs of arbitrary order. First, we propose nontrivial extensions of algorithms based on polynomial elimination and on the guess-and-prove paradigm. Second, we design two brand-new algorithms that exploit the special structure of the underlying polynomial systems. Last, but not least, we report on implementations that are able to solve highly challenging DDEs with a combinatorial origin.

翻译：离散微分方程（DDEs）是关于幂级数 $F(t,u)$ 的函数方程，该幂级数以 $t$ 为变量、以“催化”变量 $u$ 的多项式为系数，并涉及 $F(t,u)$ 及其关于 $u$ 的某些偏导数在 $u=1$ 处的特化取值。DDEs 在组合数学中出现频繁，尤其在地图枚举问题中。若 DDE 为不动点类型，则其解 $F(t,u)$ 唯一，Popescu（1986）的一般性结论表明 $F(t,u)$ 是代数幂级数。Bousquet-Mélou 与 Jehanne（2006）提出了不动点类型 DDEs 解代数性的构造性证明。Bostan 等人（2022）对一阶此类 DDEs 展开了系统性算法研究。本文将这一研究推广至任意阶 DDEs。首先，我们提出了基于多项式消元法和猜测-证明范式的算法非平凡拓展。其次，我们设计了两种全新算法，充分利用底层多项式系统的特殊结构。最后但同样重要的是，我们报告了能求解源于组合数学的高难度 DDEs 的实现方案。