Academic Journal
On Abel's problem and Gauss congruences
| العنوان: | On Abel's problem and Gauss congruences |
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| المؤلفون: | Delaygue, Eric, Rivoal, Tanguy |
| المساهمون: | Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Combinatoire, théorie des nombres (CTN), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), ANR-19-CE40-0018,DeRerumNatura,Décider l'irrationalité et la transcendance(2019), European Project: 648132,H2020,ERC-2014-CoG,ANT(2015) |
| المصدر: | ISSN: 1073-7928. |
| بيانات النشر: | CCSD Oxford University Press (OUP) |
| سنة النشر: | 2024 |
| المجموعة: | Université Jean Monnet – Saint-Etienne: HAL |
| مصطلحات موضوعية: | [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] |
| الوصف: | International audience ; A classical problem due to Abel is to determine if a differential equation $y'=\eta y$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $\eta$ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $\eta$, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $\eta$ admits a Puiseux expansion with {\em rational} coefficients at some point in $\mathbb C\cup \{\infty\}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y'=\eta y$ if and only if the coefficients of the Puiseux expansion of $x\eta(x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y'=\eta y$ with an algebraic solution when $x\eta(x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present three other applications, in particular to diagonals of rational fractions and to directed two-dimensional walks. |
| نوع الوثيقة: | article in journal/newspaper |
| اللغة: | English |
| Relation: | info:eu-repo/grantAgreement//648132/EU/Automata in Number Theory/ANT |
| DOI: | 10.1093/imrn/rnad229 |
| الاتاحة: | https://hal.science/hal-03772016 https://hal.science/hal-03772016v2/document https://hal.science/hal-03772016v2/file/abelgauss.pdf https://doi.org/10.1093/imrn/rnad229 |
| Rights: | info:eu-repo/semantics/OpenAccess |
| رقم الانضمام: | edsbas.A696144F |
| قاعدة البيانات: | BASE |
| DOI: | 10.1093/imrn/rnad229 |
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