Calculating Galois groups of third-order linear differential equations with parameters

Abstract

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel's equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In the present paper, we give the first known algorithm that calculates the differential Galois group of a third order parameterized linear differential equation.

Spectral Curves for Third-Order ODOs

“…We intend to follow our previously successful strategy, in the case of second-order Schrödinger operators [8]. In analogy with the Galois theory of univariate polynomials, a full factorization of L − λ will be studied, in combination with ideas coming from the parametric Picard-Vessiot theory [9][10][11], that must be adapted to a non-free algebraic parameter λ. Through this journey, spectral curves, planar or not, will govern the factorization of L − λ and the hidden free parameters will emerge.…”

“…Observe that whenever the spectral curve Γ is a rational curve with rational parameterization ℵ(τ) = (ℵ 1 (τ), ℵ 2 (τ), ℵ 3 (τ)), as in the previous example, one can consider the factorization of L − ℵ 1 (τ) as a differential operator in Σ(τ)[∂], since the algebraic variable τ with ∂(τ) = 0, would be a free parameter over Σ. To achieve a full factorization, it is a future project to use the parameterized Picard-Vessiot theory introduced by Cassidy and Singer in [9], and studied in [11]. It would be interesting to explore how this theory explains the factorization over the field of the spectral curve, using the results on second-order operators in [10] combined with the intrinsic factorization of Section 7.2.…”

“…We can say that it is a Picard-Vessiot theory for commuting differential operators and, consequently, λ is governed by a spectral curve. Parametric Galoisian theories, for free algebraic parameters, have existed since the pioneer parametric Picard-Vessiot theory of Cassidy and Singer [9], and have been studied for second-order operators in [10], and for third-order operators in [11], but their connection with spectral problems has not been established thus far.…”

“…It remains as a future project to define and prove the existence of the spectral Picard-Vessiot field of L − λ, for an algebro-geometric differential operator L. It would be a differential field extension of Σ(Γ), the minimal extension containing all the solutions, and it requires a full factorization of L − λ over Σ(Γ). This requires further investigations where the parametric Picard-Vessiot theory, introduced by Cassidy and Singer in [9] and studied in [10,11], may be relevant.…”

Almost-simple affine difference algebraic groups

“…In this Galois theory the Galois groups are differential algebraic groups and in this capacity they measure the differential algebraic relations (with respect to an auxiliary derivation) among the solutions of linear differential or difference equations. The structure theory of differential algebraic groups has facilitated the development of very strong hypertranscendence criteria that have been applied to various special functions ( [Arr13], [DV12], [HO15] [DHR18], [HMO17], [Har16], [AS17], [ADR], [ADH20]) and the development of algorithms for computing these Galois groups ( [Dre14], [Arr14], [Arr16], [Arr17], [MOS14], [MOS15], [MO18]).…”