Riccati differential equation for hypergeometric differential equation
type: Item
instance of: scholarly article
title: Riccati differential equation for hypergeometric differential equation
zbMATH Open document ID: 1278.12005
DOI: 10.4171/RSMUP/127-9
author: Takahiro Nakagawa
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2012-08-03T00:00:00Z
full work available at URL: http://rendiconti.math.unipd.it/volumes/vol127.php
Mathematics Subject Classification ID: 12H25
Mathematics Subject Classification ID: 12H20
Mathematics Subject Classification ID: 11S80
zbMATH DE Number: 6063122
zbMATH Keywords: Grothendieck conjecture
zbMATH Keywords: algebraic solution to differential equations
zbMATH Keywords: Riccati differential equations
zbMATH Keywords: hypergeometric differential equations
Wikidata QID: Q115211808
MaRDI profile type: Publication
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review text: The motivation of this paper is to understand whether Grothendieck's Conjecture on the existence of an algebraic solution to linear differential equations continues to hold for nonlinear differential equations. It seems that the general answer is negative, with a counterexample exhibited by \textit{M. van der Put} [Indag. Math., New Ser. 7, No. 3, 367--387 (1996; Zbl 0865.12007)]. This paper provides yet another set of negative examples coming from Riccati differential equations. NEWLINENEWLINETo be more precise, the author shows that, if \(L\) is a monic (linear) differential operator with coefficients in \(\mathbb Q(t)\) with no algebraic solution and assume that, for almost all primes \(p\), \(L\) is solvable at the generic disc \(D(t,1)\) and has one-dimensional bounded solution, then the nonlinear Riccati differential equation associated to \(L\) has no algebraic solution but has a solution algebraic over \(\mathbb F_p(t)\) for almost all \(p\). The author also studies the case of hypergeometric differential equations in detail, giving a criterion for the existence of an \(\mathbb F_p(t)\)-algebraic solution to the associated Riccati differential equation. The proof of the main theorem is built on the interplay between the solutions to the differential operator \(L\) and the solutions to the associated Riccati differential equations.
description: scientific article; zbMATH DE number 6063122