Algebraicity and integrality of solutions to differential equations
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913661709189120 |
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| author | Lam, Yeuk Hay Joshua Litt, Daniel |
| author_facet | Lam, Yeuk Hay Joshua Litt, Daniel |
| contents | We formulate a conjecture classifying algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the primes appearing in the denominators of the coefficients of their Taylor expansion at a non-singular point. For linear differential equations, this conjecture is a strengthening of the Grothendieck-Katz $p$-curvature conjecture. We prove the conjecture for many differential equations and initial conditions of algebro-geometric interest. For linear differential equations, we prove it for Picard-Fuchs equations at initial conditions corresponding to cycle classes, among other cases. For non-linear differential equations, we prove it for isomonodromy differential equations, such as the Painlevé VI equation and Schlesinger system, at initial conditions corresponding to Picard-Fuchs equations. We draw a number of algebro-geometric consequences from the proofs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13175 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraicity and integrality of solutions to differential equations Lam, Yeuk Hay Joshua Litt, Daniel Algebraic Geometry Classical Analysis and ODEs Number Theory 14G99 (Primary), 11G99 (Secondary) We formulate a conjecture classifying algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the primes appearing in the denominators of the coefficients of their Taylor expansion at a non-singular point. For linear differential equations, this conjecture is a strengthening of the Grothendieck-Katz $p$-curvature conjecture. We prove the conjecture for many differential equations and initial conditions of algebro-geometric interest. For linear differential equations, we prove it for Picard-Fuchs equations at initial conditions corresponding to cycle classes, among other cases. For non-linear differential equations, we prove it for isomonodromy differential equations, such as the Painlevé VI equation and Schlesinger system, at initial conditions corresponding to Picard-Fuchs equations. We draw a number of algebro-geometric consequences from the proofs. |
| title | Algebraicity and integrality of solutions to differential equations |
| topic | Algebraic Geometry Classical Analysis and ODEs Number Theory 14G99 (Primary), 11G99 (Secondary) |
| url | https://arxiv.org/abs/2501.13175 |