We develop algorithms to compute the differential Galois group corresponding to a one-parameter family of second order homogeneous ordinary linear differential equations with rational function coefficients. More precisely, we consider equations of the formwhere r 1 , r 2 ∈ C(x,t) and C is an algebraically closed field of characteristic zero. We work in the setting of parameterized Picard-Vessiot theory, which attaches a linear differential algebraic group to such an equation, that is, a group of invertible matrices whose entries satisfy a system of polynomial differential equations, with respect to the derivation in the parameter-space. We will compute the ∂ ∂t -differential-polynomial equations that define the corresponding parameterized Picard-Vessiot group as a differential algebraic subgroup of GL 2 .
We give simple necessary and sufficient conditions for the ∂ ∂t -transcendence of the solutions to a parameterized second order linear differential equation of the formwhere p ∈ F(x) is a rational function in x with coefficients in a ∂ ∂t -field F. This result is crucial for the development of an efficient algorithm to compute the parameterized Picard-Vessiot group of an arbitrary parameterized second-order linear differential equation over F(x). Our criteria imply, in particular, the ∂ ∂t -transcendence of the incomplete Gamma function γ(t, x), generalizing a result of [9].
Abstract. We propose a new method to compute the unipotent radical Ru(H) of the differential Galois group H associated to a parameterized second-order homogeneous linear differential equation of the formis a rational function in x with coefficients in a Π-field F of characteristic zero, and Π is a commuting set of parametric derivations. The procedure developed by Dreyfus reduces the computation of Ru(H) to solving a creative telescoping problem, whose effective solution requires the assumption that the maximal reductive quotient H/Ru(H) is a Π-constant linear differential algebraic group. When this condition is not satisfied, we compute a new set of parametric derivations Π ′ such that the associated differential Galois group H ′ has the property that H ′ /Ru(H ′ ) is Π ′ -constant, and such that Ru(H) is defined by the same differential equations as Ru(H ′ ). Thus the computation of Ru(H) is reduced to the effective computation of Ru(H ′ ). We expect that an elaboration of this method will be successful in extending the applicability of some recent algorithms developed by Minchenko, Ovchinnikov, and Singer to compute unipotent radicals for higher order equations.
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-...
Abstract. We consider first-order linear difference systems over C(x), with respect to a difference operator σ that is either a shift σ : x → x + 1, q-dilation σ : x → qx with q ∈ C × not a root of unity, or Mahler operator σ : x → x q with q ∈ Z ≥2 . Such a system is integrable if its solutions also satisfy a linear differential system; it is projectively integrable if it becomes integrable "after moding out by scalars." We apply recent results of Schäfke and Singer to characterize which groups can occur as Galois groups of integrable or projectively integrable linear difference systems. In particular, such groups must be solvable. Finally, we give hypertranscendence criteria.
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