Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation

Abstract

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.

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

“…x Y − qY = 0, whose PPV-group H is already known [3,6]. In §3, we reinterpret this change-of-variables procedure in terms of a lattice (8) of PPV-fields.…”

“…This strategy for computing G was already sketched in [1, §3.4], in the setting of one parametric derivation. The computation of the unipotent radical Ru(H) obtained in [3] for several parametric derivations enables the present generalization of [1, §3.4].…”

“…where q ∈ K. In [6], Dreyfus develops the following procedure to compute the PPV-group H corresponding to (3) (see also [1,3]). …”

“…Building on this work, unconditional algorithms to compute G are given in [1] in the setting of one parametric derivation, and in [3] for several parametric derivations but still assuming that r1 = 0. Algorithms for higher-order equations are developed in [17,18].…”

“…Algorithms for higher-order equations are developed in [17,18]. After performing a change of variables on (2), we obtain an associated equation (3) of the form δ 2…”