Burchnall-Chaundy polynomials for matrix ODOs and Picard-Vessiot Theory
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| Format: | Preprint |
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2022
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| author | Previato, Emma Rueda, Sonia L. Zurro, Maria-Angeles |
| author_facet | Previato, Emma Rueda, Sonia L. Zurro, Maria-Angeles |
| contents | Burchnall and Chaundy showed that if two ODOs $P$, $Q$ with analytic coefficients commute 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. 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. A new differential elimination tool is defined, the matrix differential resultant. It is used to compute the BC-polynomial $f$ of a pair of commuting MODOs and proved to have 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_02788 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Burchnall-Chaundy polynomials for matrix ODOs and Picard-Vessiot Theory Previato, Emma Rueda, Sonia L. Zurro, Maria-Angeles Algebraic Geometry Classical Analysis and ODEs 13N10, 13P15, 14H70 I.1.2 Burchnall and Chaundy showed that if two ODOs $P$, $Q$ with analytic coefficients commute 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. 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. A new differential elimination tool is defined, the matrix differential resultant. It is used to compute the BC-polynomial $f$ of a pair of commuting MODOs and proved to have 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. |
| title | Burchnall-Chaundy polynomials for matrix ODOs and Picard-Vessiot Theory |
| topic | Algebraic Geometry Classical Analysis and ODEs 13N10, 13P15, 14H70 I.1.2 |
| url | https://arxiv.org/abs/2210.02788 |