Burchnall–Chaundy polynomials for matrix ODOs and Picard–Vessiot theory
Burchnall and Chaundy showed that if two ordinary differential operators (ODOs) P, Q with analytic coefficients commute then there exists a polynomial f(λ, μ) with complex coefficients such that f(P, Q) = 0, called the BC-polynomial. This polynomial can be computed using the differential resultant f...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/708613 |
| Acceso en línea: | http://hdl.handle.net/10486/708613 https://dx.doi.org/10.1016/j.physd.2023.133811 |
| Access Level: | acceso abierto |
| Palabra clave: | Differential Equations Mathematical Operators Polynomials Matemáticas |
| Sumario: | Burchnall and Chaundy showed that if two ordinary differential operators (ODOs) P, Q with analytic coefficients commute then there exists a polynomial f(λ, μ) with complex coefficients such that f(P, Q) = 0, called the BC-polynomial. This polynomial can be computed using the differential resultant for ODOs. In this work we extend this result to matrix ordinary differential operators, MODOs. Our matrices have entries in a differential field K, whose field of constants C is algebraically closed and of zero characteristic. We restrict to the case of order one operators P, with invertible leading coefficient. We define a new differential elimination tool, the matrix differential resultant. We use it to compute the BC-polynomial f of a pair of commuting MODOs, and we also prove that it has constant coefficients. This resultant provides the necessary and sufficient condition for the spectral problem PY = λY, QY = μY to have a solution. Techniques from differential algebra and Picard–Vessiot theory allow us to describe explicitly isomorphisms between commutative rings of MODOs C[P, Q] and a finite product of rings of irreducible algebraic curves |
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