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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2505.01289 |
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| _version_ | 1866908346512048128 |
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| author | Jiménez-Pastor, Antonio Rueda, Sonia L. |
| author_facet | Jiménez-Pastor, Antonio Rueda, Sonia L. |
| contents | This work is devoted to computing the centralizer $Z (L)$ of an ordinary differential operator (ODO) in the ring of differential operators. Non-trivial centralizers are known to be coordinate rings of spectral curves and contain the ring of polynomials $C [L]$, with coefficients in the field of constants $C$ of $L$.
We give an algorithm to compute a basis of $Z (L)$ as a $C [L]$-module. Our approach combines results by K. Goodearl in 1985 with solving the systems of equations of the stationary Gelfand-Dickey (GD) hierarchy, which after substituting the coefficients of $L$ become linear, and whose solution sets form a flag of constants. We are assuming that the coefficients of $L$ belong to a differential algebraic extension $K$ of $C$. In addition, by considering parametric coefficients we develop an algorithm to generate families of ODOs with non trivial centralizer, in particular algebro-geometric, whose coefficients are solutions in $K$ of systems of the stationary GD hierarchy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01289 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Effective computation of centralizers of ODOs Jiménez-Pastor, Antonio Rueda, Sonia L. Rings and Algebras Algebraic Geometry 13N10, 13P15, 12H05 I.1.2 This work is devoted to computing the centralizer $Z (L)$ of an ordinary differential operator (ODO) in the ring of differential operators. Non-trivial centralizers are known to be coordinate rings of spectral curves and contain the ring of polynomials $C [L]$, with coefficients in the field of constants $C$ of $L$. We give an algorithm to compute a basis of $Z (L)$ as a $C [L]$-module. Our approach combines results by K. Goodearl in 1985 with solving the systems of equations of the stationary Gelfand-Dickey (GD) hierarchy, which after substituting the coefficients of $L$ become linear, and whose solution sets form a flag of constants. We are assuming that the coefficients of $L$ belong to a differential algebraic extension $K$ of $C$. In addition, by considering parametric coefficients we develop an algorithm to generate families of ODOs with non trivial centralizer, in particular algebro-geometric, whose coefficients are solutions in $K$ of systems of the stationary GD hierarchy. |
| title | Effective computation of centralizers of ODOs |
| topic | Rings and Algebras Algebraic Geometry 13N10, 13P15, 12H05 I.1.2 |
| url | https://arxiv.org/abs/2505.01289 |