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

Abstract

We apply the difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous linear difference equation of the form y(x + 2) + a(x)y(x + 1) + b(x)y(x) = 0, where the coefficients a(x), b(x) ∈Q(x) are rational functions in x with coefficients inQ. We develop algorithms to compute the differencedifferential Galois group associated to such an equation, and show how to deduce the differentialalgebraic relations among the solutions from the defining equations of the Galois group. IntroductionConsider a second-order homogeneous linear difference equation σ 2 (y) + aσ(y) + by = 0, (1.1) whose coefficients a, b ∈Q(x), and where σ denotes theQ-linear automorphism defined by σ(x) = x + 1. We are motivated by the question: do the solutions of (1.1) satisfy any d dx -algebraic equations overQ(x)? And if so, how can we compute all such differential-algebraic relations? We give complete answers to these questions as an application of the difference-differential Galois theory developed in [21], which studies equations such as (1.1) from a purely algebraic point of view. This theory attaches a linear differential algebraic group G (Definition 2.7) to (1.1), which group encodes all the difference-differential algebraic relations among the solutions to (1.1). We develop an algorithm to compute G, and then show how the knowledge of G leads to a concrete description of the sought difference-differential algebraic relations among the solutions. The difference-differential Galois theory of [21] is a generalization of the difference Galois theory presented in [38], where the Galois groups that arise are linear algebraic groups that encode the difference-algebraic relations among the solutions to a given linear difference equation. An algorithm to compute the difference Galois group H associated to (1.1) by the theory of [38] is developed in [23]. The computation of G is more difficult than that of H, because there are many more linear differential algebraic groups than there are linear algebraic groups (more precisely, the latter are instances of the former), so identifying the correct differencedifferential Galois group from among these possibilities requires additional work. 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-...

Galois groups for integrable and projectively integrable linear difference equations

“…4 All fields considered in this paper are of characteristic zero. and H be the σ-Galois group of E over C. The map sending γ ∈ G to its restriction γ| E to E is an isomorphism of G onto the σ-Galois group of E over E ∩ k, which latter group is a subgroup of H. The result now follows from (1).…”

“…] is a σ 2 -Picard-Vessiot ring for (4.23) over the underlying σ 2 -field of k. Using the algorithm of [15], we see that S (0) and S (1) are domains, which implies that t = 2, S (0) = S 0 , and S (1) = S 1 .…”

“…In general, t is the largest integer such that there exists z ∈ S with σ(z) = ζ t z, where ζ t is a primitive t-th root of unity. Since σ(z 1 /z 2 ) = −z 1 /z 2 , we have that t is even, and therefore S = S (0) ⊕ S (1) , where each of…”

“…Proof. Let K be the total quotient ring of a σδ-Picard-Vessiot extension R = k{Z, 1 det Z } δ of k corresponding to (5.1), and let G be its σδ-Galois group over k. Then S = k[Z, 1 det Z ] is a σ-Picard-Vessiot extension for this equation, and its total quotient ring F is a subring of K [11,Cor. 20].…”

“…Besides applications to questions of hypertranscendence, our results are also useful in understanding the inverse problem for the differential Galois theory of difference equations by showing that certain groups cannot occur as differential Galois groups. These results have also been applied in [1] to develop algorithms that compute differential Galois groups of difference equations.…”