D-Algebraic Functions

TL;DR

This paper investigates the properties of D-algebraic functions, providing constructive proofs of their closure properties, algorithms for manipulating them, and applications to scientific examples.

Contribution

It introduces algorithms for computing differential equations of D-algebraic functions and establishes bounds on their order, enhancing understanding of their algebraic structure.

Findings

Algorithms successfully compute differential equations for function compositions.

Bounds on the order of resulting differential equations are derived.

Applications demonstrate practical utility in scientific contexts.

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.