Solving second order linear differential equations with Klein's theorem

Hoeij, Mark van; Weil, Jacques-Arthur

doi:10.1145/1073884.1073931

Cited by 26 publications

(35 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If 1/k + 1/ + 1/m > 1 and n ≥ 3, there are (k, , m)-minus-n Belyi functions of arbitrary high degree. They give Kleinian pull-back transformations [13,25] to second order Fuchsian equations with finite *…”

mentioning

confidence: 99%

Belyi functions for hyperbolic hypergeometric-to-Heun transformations

Hoeij

Vidūnas

2015

Journal of Algebra

View full text Add to dashboard Cite

A complete classification of Belyi functions for transforming certain hypergeometric equations to Heun equations is given. The considered hypergeometric equations have the local exponent differences 1/k, 1/ , 1/m that satisfy k, , m ∈ N and the hyperbolic condition 1/k + 1/ + 1/m < 1. There are 366 Galois orbits of Belyi functions giving the considered (non-parametric) hypergeometric-to-Heun pull-back transformations. Their maximal degree is 60, which is well beyond reach of standard computational methods. To obtain these Belyi functions, we developed two efficient algorithms that exploit the implied pull-back transformations.

show abstract

mentioning

confidence: 99%

Belyi functions for hyperbolic hypergeometric-to-Heun transformations

Hoeij

Vidūnas

2015

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…For each integer n ∈ N * , it admits a rational first integral of degree N = 4 n + 1. This system is derived from the Riccati equation of a standard hypergeometric equation with a finite dihedral differential Galois group, see [vHW05]. The following table contains the timings (in seconds) for HeuristicRationalFirstIntegral to find a rational first integral of degree N = 4 n + 1 when it is run with N = 4 n + 1.…”

Section: Implementation and Experimentsmentioning

confidence: 99%

Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields

et al. 2015

View full text Add to dashboard Cite

Abstract. We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach is inspired by an idea of Ferragut and Giacomini ([FG10]). We improve upon their work by proving that rational first integrals can be computed via systems of linear equations instead of systems of quadratic equations. The main ingredients of our algorithms are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating a power series. This leads to a probabilistic algorithm with arithmetic complexityÕ(N 2ω ) and to a deterministic algorithm solving the problem iñ O(d 2 N 2ω+1 ) arithmetic operations, where N denotes the given bound for the degree of the rational first integral, and where d ≤ N is the degree of the vector field, and ω the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, inÕ(N ω+2 ) arithmetic operations. By comparison, the best previous algorithm given in [Chè11] uses at least d ω+1 N 4ω+4 arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package RationalFirstIntegrals which is available to interested readers with examples showing its efficiency.

show abstract

“…If there exists Liouvillian solutions, then return them. The algorithm in [10] computes Liouvillian solutions in form (1) if Linp is irreducible.…”

Section: Outputmentioning

confidence: 99%