Algebraic solutions of linear differential equations: an arithmetic approach
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914921062596608 |
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| author | Bostan, Alin Caruso, Xavier Roques, Julien |
| author_facet | Bostan, Alin Caruso, Xavier Roques, Julien |
| contents | Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz $p$-curvature conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_05061 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Algebraic solutions of linear differential equations: an arithmetic approach Bostan, Alin Caruso, Xavier Roques, Julien Number Theory Symbolic Computation Classical Analysis and ODEs Combinatorics Primary: 11-02, Secondary: 12H05, 33C20. 12H25, 34A20, 34A30, 34M15, 05A15, 68W30 Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz $p$-curvature conjecture. |
| title | Algebraic solutions of linear differential equations: an arithmetic approach |
| topic | Number Theory Symbolic Computation Classical Analysis and ODEs Combinatorics Primary: 11-02, Secondary: 12H05, 33C20. 12H25, 34A20, 34A30, 34M15, 05A15, 68W30 |
| url | https://arxiv.org/abs/2304.05061 |