  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.
