Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

Abstract

The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman and M.F. Singer presented an algorithm calculating the differential Galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators. We use their algorithm as a part of our algorithm. As a result, we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter.

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

“…Let r stand for the Lie algebra of R u (G) (see [16,Section 4.1] for a quick and sufficient overview) and Z be defined by (3.4). Then, under the matrix conjugation, Therefore, if [G, G] is non-commutative, so is R u (G).…”

“…This is the character of the action of G on Lie Z by conjugation. Since Lie G ∩ Lie Z is G-invariant, [16,Proposition 3.21] implies that Lie G ∩ LieZ is a k-subspace of Lie Z, and so either B, B], which is not commutative. Since Lie R u (G)/ Lie Z contains A, it therefore coincides with A (all non-zero proper k-subspaces of L are maximal since dim k L = 2).…”

“…The existing methods of studying hypertranscedence of solutions of second order inhomogeneous linear differential equations [16] use results from parameterized differential Galois theory [11,27]. In this application, these results essentially limit the consideration of inhomogeneous terms to rational functions.…”

“…The hypertranscendence results of [16] are based in part on new results in the representation theory of linear differential algebraic groups. In particular, extensions of a trivial representation by a two-dimensional representation are analyzed there.…”

Differential Galois Theory and Isomonodromic Deformations

“…The new aspect of the parameterized Picard-Vessiot theory, compared to the classical one, is that the Galois group does not only encode algebraic dependence relations between the coefficients of U , but differential algebraic dependence relations as functions of the parameters. Computation of the parameterized differential Galois groups is a delicate matter, many articles have been devoted to the question among them [1,8,13,24,25].…”