Computing the differential Galois group of a parameterized second-order linear differential equation

Abstract

We develop algorithms to compute the differential Galois group G associated to a parameterized second-order homogeneous linear differential equation of the formx with coefficients in a partial differential field F of characteristic zero. This work relies on earlier procedures developed by Dreyfus and by the present author to compute G when r1 = 0. By reinterpreting a classical change-ofvariables procedure in Galois-theoretic terms, we complete these algorithms to compute G with no restrictions on r1.

Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation

“…This reduction is not available in the difference-differential Galois theory, where it is not always possible to tensor away the effect of the determinant, so we must compute G directly. An upshot of this is that the defining equations for G obtained here are more explicit than those produced in [3,4] for the parameterized differential setting.A second complication is that the inverse problem in the difference-differential Galois theory (that is: which linear differential algebraic groups arise as difference-differential Galois groups?) remains open, whereas in the parameterized differential Galois theory there is a complete answer to this question proved in [10,32].…”

“…However, the difference Galois group H serves as a close upper bound for the difference-differential Galois group G: it is shown in [21] that one can consider G as a Zariski-dense subgroup of H without loss of generality (see Proposition 2.12 for a precise statement). In view of this fact, our strategy to compute G is to first apply the algorithm of [23] to compute H, and then compute the additional differential-algebraic equations (if any) that define G as a subgroup of H.This strategy is reminiscent of the one begun in [11], and concluded in [2][3][4], to compute the parameterized differential Galois group for a second-order linear differential equation with differential parameters, where the results of [6,28] are first applied to compute the classical (non-parameterized) differential Galois group for the differential equation, and one then computes the additional differential-algebraic equations, with respect to the parametric derivations, that define the parameterized differential Galois group inside the classical one. However, the computation of the difference-differential Galois group for (1.1) presents substantial new complications, which we describe below.Firstly, in the parameterized differential algorithm one first computes an associated unimodular differential Galois group as in [2,11], and then recovers the original Galois group from this associated unimodular group and the change-of-variables data as in [3,4].…”

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

“…. , y n and with coefficients in a ∆-field (K, ∆), is the ring of polynomials in the indeterminates formally denoted δ i 1 1 · . .…”

“…This gives rise to the action of P 1 × P 2 on X := X 1 × X 2 . The group P := P 1 ∩ P 2 embedded diagonally into P 1 × P 2 also acts on X with the stabilizer of the point eG 1…”

“…. , a h ∈ k be such thatψ(y) = h ∑ i=0 a i y (i) .Condition(3.15) also implies that, for all a ∈ A,(3.17) ϕ(a) = ψ(a) − aψ(1).Hence, for all a, b ∈ A, ψ(ab) = aψ(b) + bψ(a) − abψ(1).We then have, for all a, b ∈ A, (i) + ba (i) − aba 0 , (i) + ba (i) .Hence, for all a, b ∈ A, j) b (i− j) = 0.For each non-constant a ∈ A, this implies that the non-zero linear differential polynomial f (y) j) y (i− j)…”