Reduction systems and degree bounds for integration
Hao Du, Clemens G. Raab
TL;DR
The work addresses the challenge of finding elementary antiderivatives within symbolic integration by refining the Risch--Norman framework. It formalizes Norman's completion-based reduction approach, introduces a refined completion procedure that can terminate in cases where the original does not, and develops a rigorous theory of complete reduction systems. The authors also analyze infinite reduction systems arising in Airy functions and complete elliptic integrals, and derive rigorous degree bounds from these systems, including tight, weight-aware bounds. Collectively, these results enhance both the robustness and the theoretical grounding of reduction-based integration, enabling more reliable termination behavior and precise degree estimates for the numerator in the Risch--Norman ansatz. The practical impact lies in enabling stronger symbolic integration capabilities and providing a blueprint for deriving exact degree bounds in nontrivial differential fields.
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.