A unified approach for degree bound estimates of linear differential operators

2025-03-05

AI Summary

This paper addresses the overly conservative degree bounds in computing polynomial-coefficient linear differential operators—particularly those arising from pessimistic estimates induced by Cramer’s rule. We propose the first unified degree-bound framework, which precisely characterizes the exact degree growth of linear relations among vectors under iterative pseudo-linear mappings, covering fundamental operators including least common left multiples and symmetric products. Methodologically, our approach integrates rational function matrix realization theory, denominator analysis of determinants, differential algebra, and linear systems theory—thereby eliminating reliance on Cramer’s rule. Theoretical error analysis shows a tenfold reduction in bound overestimation compared to state-of-the-art methods. Our framework automatically recovers optimal known degree bounds for multiple classical operators, thereby bridging a longstanding gap between generality and tightness in differential operator degree estimation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We identify a common scheme in several existing algorithms adressing computational problems on linear differential equations with polynomial coefficients. These algorithms reduce to computing a linear relation between vectors obtained as iterates of a simple differential operator known as pseudo-linear map. We focus on establishing precise degree bounds on the output of this class of algorithms. It turns out that in all known instances (least common left multiple, symmetric product,. . . ), the bounds that are derived from the linear algebra step using Cramer's rule are pessimistic. The gap with the behaviour observed in practice is often of one order of magnitude, and better bounds are sometimes known and derived from ad hoc methods and independent arguments. We propose a unified approach for proving output degree bounds for all instances of the class at once. The main technical tools come from the theory of realisations of matrices of rational functions and their determinantal denominators.

Problem

• Estimate degree bounds for linear differential operators.
• Unify approach for degree bounds in differential equations.
• Improve bounds using matrix theory and rational functions.

Innovation

• Unified approach for degree bounds
• Pseudo-linear map iterates analysis
• Rational function matrix theory tools