Arithmetic of D-Algebraic Functions
Bertrand Teguia Tabuguia
TL;DR
This work develops a computational theory for the arithmetic of $D$-algebraic functions, i.e., zeros of algebraic differential equations in one or more variables. It introduces a unifying framework using differential ideals, triangular sets, and saturation to derive ADEs satisfied by rational expressions of known $D$-algebraic functions, and extends previous univariate results to multivariate PDEs with careful order/degree considerations. The authors show how to obtain least-order ADEs and provide a Maple-based implementation (NLDE) that handles both l.h.o. and non-l.h.o. cases, as well as radical differential ideals. They illustrate the approach with classic examples (e.g., the Weierstrass function) and discuss broad applications in physics, statistics, and symbolic integration uses. The work lays groundwork for effective, structure-preserving algebraic manipulations of $D$-algebraic functions and their applications in counting problems, integral transforms, and dynamical systems modeling.
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 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.