Computing the differential Galois group of a one-parameter family of\n second order linear differential equations

2012/08/10 by Carlos E. Arreche, Arreche, Carlos E.
Computer Science · Mathematics · Physics and Astronomy · #12H20 #13N10 #20H20 #34M03 #34M15 #37K20 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Classical Analysis and ODEs (math.CA) #Commutative Algebra (math.AC) #FOS: Mathematics #Nonlinear Waves and Solitons #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1208.2226

openalex publication_date 2012/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop algorithms to compute the differential Galois group corresponding\nto a one-parameter family of second order homogeneous ordinary linear\ndifferential equations with rational function coefficients. More precisely, we\nconsider equations of the form \(\∂²Y)/(\∂ x²)+\nr₁\(\∂ Y)/(\∂ x) +r₂Y=0, where r₁,r₂\∈ C(x,t) and C is\nan algebraically closed field of characteristic zero.\n We work in the setting of parameterized Picard-Vessiot theory, which attaches\na linear differential algebraic group to such an equation, that is, a group of\ninvertible matrices whose entries satisfy a system of polynomial differential\nequations, with respect to the derivation in the parameter-space. We will\ncompute the \(\∂)/(\∂ t)-differential-polynomial equations\nthat define the corresponding parameterized Picard-Vessiot group as a\ndifferential algebraic subgroup of \GL₂.\n

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