Degree bounds for linear differential equations and recurrences

TL;DR

This paper introduces a unified method for deriving tight degree bounds on solutions of linear differential equations and recurrences, improving upon or recovering previous bounds across various computational problems.

Contribution

It presents a general framework that provides precise, tight degree bounds for solutions, unifying and enhancing existing ad hoc methods.

Findings

The approach yields tight degree bounds for multiple classes of problems.

It improves or recovers the best known bounds for these problems.

The method is applicable to a wide range of algorithms involving pseudo-linear maps.

Abstract

Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing algorithms that compute such representations as a linear relation between the iterates of an elementary operator known as a \emph{pseudo-linear map}. Algorithms of this form have been designed and used for solving various computational problems, in different contexts, including effective closure properties for linear differential or recurrence equations, the computation of a differential equation satisfied by an algebraic function, and many others. We propose a unified approach for establishing precise degree bounds on the solutions of all these problems. This approach relies on a common structure shared by all the specific instances of the class. For each problem, the obtained bound is tight. It either improves or recovers the previous best known bound that was derived by \emph{ad hoc} methods.