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：我们研究常或偏非线性微分方程解的算术性质，这类方程关于未知函数及其导数是代数的。我们将这些解称为D-代数函数，其方程为代数（常或偏）微分方程。总体目标是寻找以给定ADE的解的指定有理表达式为解的ADE。对于单变量D-代数函数，我们展示了如何推导出最小可能阶数的ADE。在多变量情形，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。利用配套的Maple软件，我们讨论了在物理学、统计学和符号积分中的应用。