Algorithms for D-Algebraic Functions

When you are lucky, you can describe a mathematical function by an explicit expression, as for example in the case f(x)=x^3. It is quite comfortable to work with such expressions. However, most functions that arise in mathematical contexts cannot be described that easily. They are too complicated to be expressed through a single explicit formula. People are therefore looking for alternative ways to describe functions. So-called differential equations offer one such alternative way. Differential equations are equations that express a relationship between a function and its derivatives, such as for example x*f`(x) - 3*f(x) = 0. There are many functions which cannot be described by an explicit expression but by such a differential equation. Also these description are quite comfortable to work with. This is true at least for certain differential equations. In the past few decades, primarily so-called linear differential equations have been investigated in the area of computer algebra, because these are particularly convenient from a computational point of view, and they cover quite a large set of interesting functions. Sometimes however, we also encounter functions whose descriptions require more complicated types of differential equations. Such differential equations, as well as the functions that can be described by them, are in the focus of the project. We want to develop computer algebra methods that are able to automatically answer typical questions about such functions. A simple example for such a question is the following: what is the differential equation that describes the product of two given functions which are themselves described by differential equations. For this question and others, we currently know only rather rudimentary methods, or even no methods at all. This is problematic because functions described by nonlinear differential equations have been showing up more often in various parts of mathematics since recently. The methods developed in the project are therefore not only of theoretical interest, but they are also needed. The goals of the projects are therefore not limited to gaining a better understanding of how to compute with complicated functions, but it is also a goal to implement the corresponding methods in useful computer algebra software so that the developed methods can be applied in other areas of science.