Computing differential subresultants with Maple

2025-11-28

AI Summary

This paper addresses the computation of the greatest common right divisor (GCRD) for linear ordinary differential operators with non-rational coefficients—such as parametric transcendental functions—where classical algorithms assume rational-function coefficients. The proposed symbolic algorithm is grounded in the determinant definition of the differential resultant; it explicitly constructs the resultant matrix and leverages Maple’s DEtools package and symbolic computation capabilities for efficient GCRD computation, naturally accommodating parametric coefficients. Key contributions include: (i) the first implementation of a GCRD algorithm for non-rational coefficient operators within a general-purpose symbolic computation platform; (ii) a practical, parameter-aware tool for analyzing parametric differential systems; and (iii) successful application to the construction of commuting differential operators, thereby empirically validating the algorithm’s correctness and effectiveness.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We review the definition and main properties of differential subresultants in order to achieve their implementation in Maple, using the DEtools package. The focus is on computing GCRDs of ordinary differential operators with non necessarily rational coefficients. Determinant expressions provide explicit control, enabling the treatment of coefficients with parameters. Applications to commuting ordinary differential operators illustrate the effectiveness of the method.

Problem

• Implementing differential subresultants in Maple for GCRD computation
• Handling ordinary differential operators with non-rational coefficients
• Applying the method to analyze commuting differential operators

Innovation

• Implemented differential subresultants using Maple DEtools
• Computed GCRDs for operators with non-rational coefficients
• Used determinant expressions to handle parametric coefficients