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 are specified rational expressions of solutions to given ADEs. For univariate D-algebraic functions, we show how to derive an ADE whose order is bounded by the sum of the orders of the given algebraic ODEs. In the multivariate case, we prove that this cannot be done with algebraic PDEs, and introduce a general algorithm for these computations. Using our accompanying Maple software, we discuss applications in physics, statistics, and symbolic integration.

翻译：本文关注于在自变量及其导数上为代数形式的常微分或偏微分非线性微分方程的算术性质。我们将这些解称为D-代数函数，其对应的方程为代数（常或偏）微分方程。总体目标是寻找以给定ADE解的有理表达式作为解的ADE。对于单变量D-代数函数，我们展示了如何推导出一个阶数受给定代数常微分方程阶数之和约束的ADE。在多变量情形下，我们证明了无法通过代数偏微分方程实现这一目标，并为此类计算引入了一种通用算法。借助我们配套的Maple软件，讨论了在物理学、统计学和符号积分中的应用。