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）是关联形式幂级数 $F(t,u)$（其关于 $t$ 的系数为关于“催化”变量 $u$ 的多项式）与 $F(t,u)$ 在特定点（例如 $u=1$）及若干关于 $u$ 的偏导数取值之间的多项式型函数方程。此类方程在组合学（尤其是地图枚举）中频繁出现。若某DDE属于不动点类型，则其解 $F(t,u)$ 唯一；Popescu（1986）的一般性结论表明，此时 $F(t,u)$ 为代数形式幂级数。Bousquet-Mélou 与 Jehanne（2006）曾提出针对不动点类型DDE解代数性的构造性证明。Bostan等人（2022）对一阶这类DDE进行了系统性算法研究。本文将这一研究推广至任意阶DDE。首先，我们提出基于多项式消元与猜想-证明范式的算法扩展；其次，设计两种全新算法以利用底层多项式系统的特殊结构；最后，我们报告了能够求解源于组合学的极具挑战性DDE的实现方案。