Computing differential subresultants with Maple
M. Cabellos, S. L. Rueda
TL;DR
The paper extends subresultant theory to differential operators by developing differential resultants and subresultants, enabling explicit computation of greatest common right divisors (GCRDs) for ODOs with non-rational coefficients. It presents a determinant-based framework via Sylvester matrices and introduces a Maple implementation (DSres) that leverages DEtools for symbolic analysis and parametric coefficient handling. The approach is demonstrated in the context of commuting differential operators, where the Burchnall-Chaundy spectral curve emerges from the differential resultant, and the GCRD over the spectral-curve field $K(\Gamma)$ encodes the associated algebraic data. Concrete examples with Euler and Lamé operators illustrate the method’s ability to extract spectral curves and GCRDs, while identifying current limitations in non-rational settings and pointing to avenues for extending factorization over spectral curves.
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.