# Arithmetic of D-Algebraic Functions

Bertrand Teguia Tabuguia: Arithmetic of D-algebraic Functions

ABSTRACT: 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.

For instance, the square of the probability density function

$\begin{equation*} f(x,\mu) := \frac{1}{\sigma \sqrt{2\,\pi}}\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right), \end{equation*}$

of the univariate normal distribution, seen as a bivariate function in the mean and the indeterminate variable, satisfies the following algebraic PDE (see Example 14 in the article)

$\begin{equation*} -z \! \left(x ,\mu \right) \left(\frac{\partial^{2}}{\partial \mu \partial x}z \! \left(x ,\mu \right)\right) \sigma^{2}+\left(\frac{\partial}{\partial \mu}z \! \left(x ,\mu \right)\right) \left(\frac{\partial}{\partial x}z \! \left(x ,\mu \right)\right) \sigma^{2}+2 z \! \left(x ,\mu \right)^{2}=0. \end{equation*}$

This work comes as a complement of D-Algebraic Functions with an extension to the arithmetic of multivariate D-algebraic functions. The NLDE package contains a subpackage called MultiDalg whose current main feature is arithmeticMDalg to work with algebraic PDEs. A version of the NLDE package is directly made available here NLDE.mla. For the most recent version and source code visit Github NLDE.

The following file is a Maple worksheet with the computations of the article.

The outputs are presented in the following printed pdf:

Project page created: 01/05/2023

Project contributors: Bertrand Teguia Tabuguia