On Solutions of Linear Functional Systems and Factorization of Laurent–ore Modules

Abstract

We summarize some recent results on partial linear functional systems. By associating a finite-dimensional linear functional system to a Laurent-Ore module, Picard-Vessiot extensions are generalized from linear ordinary differential (difference) equations to finite-dimensional linear functional systems. A generalized Beke's method is also presented for factoring Laurent-Ore modules and it will allow us to find all "subsystems"whose solution spaces are contained in that of a given linear functional system.

Differential Galois theory of linear difference equations

“…Note that the system σ(B) = ABA −1 + ∂(A)A −1 is an inhomogeneous system of linear difference equations in the entries of B and that there are well developed algorithms to determine if such a system has a solution whose entries are rational functions, [2,5,7,21,22,35,50]. We will give two examples to illustrate ad hoc strategies.…”

“…-Π is empty, one is considering the differential/difference fields and linear functional equations of [50] and [8].…”

Determining Whether a Multivariate Hyperexponential Function is Algebraic

“…By Proposition 5.2, r j = ω j lσ j (f j ) for all j with + 1 ≤ j ≤ m. The equality (11) implies that…”

“…, x ). If (10), (11) and (12) hold, and (9) has a minimal solution (t, f ), then f = qg t where q ∈ F and g ∈ F .…”

“…They play important roles in factoring modules over Laurent-Ore algebras, and in the continuous and discrete versions of Zeilberger's algorithm for proving combinatorial identities (see [11] and [7], respectively).…”