Reduction systems and degree bounds for integration

TL;DR

This paper formalizes Norman’s reduction approach in symbolic integration, enhancing termination detection and applying it to complex functions like Airy and elliptic integrals, while also deriving precise degree bounds.

Contribution

It formalizes Norman’s reduction system approach, develops a refined completion process, and applies it to complex functions to obtain tight degree bounds.

Findings

Refined completion process terminates in more cases.

Infinite reduction systems can be described for special functions.

General formula for weighted degree bounds is provided.

Abstract

In symbolic integration, the Risch--Norman algorithm aims to find closed forms of elementary integrals over differential fields by an ansatz for the integral, which usually is based on heuristic degree bounds. Norman presented an approach that avoids degree bounds and only relies on the completion of reduction systems. We give a formalization of his approach and we develop a refined completion process, which terminates in more instances. In some situations when the completion process does not terminate, one can detect patterns allowing to still describe infinite reduction systems that are complete. We present such infinite systems for the fields generated by Airy functions and complete elliptic integrals, respectively. Moreover, we show how complete reduction systems can be used to find rigorous degree bounds. In particular, we give a general formula for weighted degree bounds and we apply it to find tight bounds in the above examples.