Arithmetic of D-Algebraic Functions

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.