# D-Algebraic Functions¶

Rida Ait El Manssour, Anna-Laura Sattelberger, and Bertrand Teguia Tabuguia: D-Algebraic Functions

ABSTRACT: Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We present algorithms to compute algebraic differential equations for compositions and arithmetic manipulations of univariate D-algebraic functions and derive bounds for the order of the resulting differential equations. We apply our methods to examples in the sciences.

For instance, the exponential of the Painlevé transcendent of type I is a D-algebraic function. It fullfils the following ADE (see Example 5.7 in our article):

\begin{split}\begin{align*} \hspace{-11mm} {24 x {w'(x)}^{2} w \! \left(x \right)^4+w \! \left(x \right)^{6}-2 w \! \left(x \right)^{5} w^{(3)}(x)+6 w''(x) w'(x) w \! \left(x \right)^4}+{w^{(3)}(x)}^{2} w \! \left(x \right)^4 \\ \hspace{-11mm} -4 {w'(x)}^3-24 w''(x) {w'(x)}^{2} w \! \left(x \right)^3-6 w^{(3)}(x) w''(x) w'(x) w \! \left(x \right)^3+24 {w'(x)}^4 w \! \left(x \right)^{2}\\ \hspace{-11mm} {+4 w^{(3)}(x) w'(x)^3 w \! \left(x \right)^{2}+9 {w''(x)}^{2} {w'(x)}^{2} w \! \left(x \right)^{2}}-12 w''(x) {w'(x)}^4 w \! \left(x \right)+4 {w'(x)}^{6}\,=\, 0 \,. \hspace{-11mm} \end{align*}\end{split}

We have described two methods of computations, and each of them yields algorithms. The algorithms of our second method are implemented in Maple in the package NLDE (NonLinear algebra and Differential Equations). The package can be downloaded here NLDE.mla, and its source here sourceNLDE.mpl. This is a user-friendly interface to the code for any Maple user who wants to try our implelentation. Questions and comments are welcome. The use of the package is illustrated in the following notebook:

The following file is a Maple worksheet with the same computations.

The outputs are presented in the following printed pdf:

We also have a Macaulay2 code that implements our first method. The following notebook demonstrates how it can be used:

Project page created: 14/11/2022

Project contributors: Rida Ait El Manssour, Anna-Laura Sattelberger, Bertrand Teguia Tabuguia