Border Bases in the Rational Weyl Algebra
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911442927616000 |
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| author | Rodriguez, Carlos Sattelberger, Anna-Laura |
| author_facet | Rodriguez, Carlos Sattelberger, Anna-Laura |
| contents | Border bases are a generalization of Gröbner bases for zero-dimensional ideals in polynomial rings. In this article, we introduce border bases for a non-commutative ring of linear differential operators, namely the rational Weyl algebra. We elaborate on their properties and present algorithms to compute with them. We apply this theory to represent integrable connections as cyclic $D$-modules explicitly. As an application, we visit differential equations behind a string, a Feynman as well as a cosmological integral. We also address the classification of particular $D$-ideals of a fixed holonomic rank, namely the case of linear PDEs with constant coefficients as well as Frobenius ideals. Our approach rests on the theory of Hilbert schemes of points in affine space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23411 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Border Bases in the Rational Weyl Algebra Rodriguez, Carlos Sattelberger, Anna-Laura Algebraic Geometry Symbolic Computation High Energy Physics - Theory Border bases are a generalization of Gröbner bases for zero-dimensional ideals in polynomial rings. In this article, we introduce border bases for a non-commutative ring of linear differential operators, namely the rational Weyl algebra. We elaborate on their properties and present algorithms to compute with them. We apply this theory to represent integrable connections as cyclic $D$-modules explicitly. As an application, we visit differential equations behind a string, a Feynman as well as a cosmological integral. We also address the classification of particular $D$-ideals of a fixed holonomic rank, namely the case of linear PDEs with constant coefficients as well as Frobenius ideals. Our approach rests on the theory of Hilbert schemes of points in affine space. |
| title | Border Bases in the Rational Weyl Algebra |
| topic | Algebraic Geometry Symbolic Computation High Energy Physics - Theory |
| url | https://arxiv.org/abs/2510.23411 |