Degree bounds for linear differential equations and recurrences

2026-01-13

AI Summary

This work establishes tight degree bounds for solutions of linear differential equations and recurrence relations to enable efficient representation and computation of special functions. By introducing a unified algorithmic framework based on iterated pseudolinear maps, the approach systematically addresses a range of problems—including closure properties and solving differential equations for algebraic functions—overcoming the limitations of previous ad hoc methods. Within this general structure, the authors derive, for the first time, uniform and tight degree bounds that not only recover known optimal results but also improve upon them in several cases, thereby significantly enhancing both theoretical clarity and computational practicality.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Other Foundations of Machine Learning

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• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• degree bounds
• linear differential equations
• recurrences
• pseudo-linear maps
• special functions

Innovation

• pseudo-linear map
• degree bounds
• linear differential equations
• recurrence relations
• unified approach