We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations are algebraic (ordinary or partial) differential equations (ADEs). The general purpose is to find ADEs whose solutions contain specified rational expressions of solutions to given ADEs. For univariate D-algebraic functions, we show how to derive an ADE of smallest possible order. In the multivariate case, we introduce a general algorithm for these computations and derive conclusions on the order bound of the resulting algebraic PDE. Using our accompanying Maple software, we discuss applications in physics, statistics, and symbolic integration.

翻译：本文研究常系数或偏系数非线性微分方程解的算术性质，这些方程在未定元及其导数上是代数的。我们称此类解为D-代数函数，其对应的方程为代数（常系数或偏系数）微分方程（ADEs）。本文的核心目标是寻找这样的ADEs：其解包含给定ADEs解的特定有理表达式。对于单变量D-代数函数，我们展示了如何推导出可能最低阶的ADE。在多变量情形中，我们引入了一种通用算法用于此类计算，并推导了所得代数偏微分方程阶数上界的相关结论。借助我们配套的Maple软件，我们讨论了该方法在物理学、统计学及符号积分领域的应用。