In this article, we study systems of $n \geq 1$, not necessarily linear, discrete differential equations (DDEs) of order $k \geq 1$ with one catalytic variable. We provide a constructive and elementary proof of algebraicity of the solutions of such equations. This part of the present article can be seen as a generalization of the pioneering work by Bousquet-M\'elou and Jehanne~(2006) who settled down the case $n=1$. Moreover, we obtain effective bounds for the algebraicity degrees of the solutions and provide an algorithm for computing annihilating polynomials of the algebraic series. Finally, we carry out a first analysis in the direction of effectivity for solving systems of DDEs in view of practical applications.

翻译：本文研究含一个催化变量的$n \geq 1$个（未必线性）$k \geq 1$阶离散微分方程（DDEs）系统。我们提供了此类方程解的代数性质的构造性初等证明。本文这一部分可视为Bousquet-Mélou与Jehanne（2006年）奠基性工作的推广，后者仅解决了$n=1$的情形。此外，我们获得了解代数次数的有效界，并给出了计算代数级数消去多项式的算法。最后，面向实际应用，我们针对DDEs系统求解的有效性问题进行了初步分析。