Riccati differential equation for hypergeometric differential equation
algebraic solution to differential equations
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.
scientific article; zbMATH DE number 6063122
scientific article