Problem
Research questions and friction points this paper is trying to address.
linear ordinary differential operators
operator arithmetic
factorization
differential equations
computer algebra
Algorithms for Linear Ordinary Differential Operators
🤖 AI Summary
This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.
📝 Abstract
We describe an implementation in Scratchpad II of linear ordinary differential operators over a differential ring, acting on a module equipped with a compatible derivation. The abstract data type facilities of the system allow such operators to be represented and manipulated as first-class objects while retaining the usual notation for operator application. For coefficients in a field, we give constructive algorithms for left and right division, greatest common divisors, least common multiples, and an extended Euclidean algorithm; the right-hand constructions may be obtained from the corresponding left-hand constructions in the opposite ring. We also discuss pseudo-division over polynomial coefficient rings and an Ore localization yielding a right field of fractions. Finally, we apply this operator arithmetic to factorization of ordinary differential equations, using the associated Riccati equation and Newton polygons to analyze possible singular parts of factors. Examples include operators with constant, elementary-function, rational-function, and matrix coefficients.
Innovation
Methods, ideas, or system contributions that make the work stand out.
linear ordinary differential operators
operator arithmetic
Ore localization
factorization of differential equations
extended Euclidean algorithm
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