Picard--Vessiot extensions for linear functional systems

Abstract

Picard-Vessiot extensions for ordinary differential and difference equations are well known and are at the core of the associated Galois theories. In this paper, we construct fundamental matrices and Picard-Vessiot extensions for systems of linear partial functional equations having finite linear dimension. We then use those extensions to show that all the solutions of a factor of such a system can be completed to solutions of the original system.

Differential Galois theory of linear difference equations

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

Testing linear dependence of hyperexponential elements

“…Thus, k x 1 3 x k is in some ∆-extension of C(x, k) (see [1,Theorem 1]), and is hyperexponential over C(x, k).…”

A recursive method for determining the one-dimensional submodules of Laurent-Ore modules

“…, ∂ −1 m ] whose multiplication rules are ∂s∂t = ∂t∂s, ∂j∂ −1 j = 1, ∂ia = a∂i+δi(a), ∂ja = σj(a)∂j, and ∂ −1 j a = σ −1 j (a)∂ −1 j , where 1 ≤ s < t ≤ m, 1 ≤ i≤ , + 1 ≤ j ≤ m, and a ∈ F . The algebra L can be constructed from an Ore algebra over F (see [7]). For any finite-dimensional L-module, its F -bases may be computed via the Gröbner basis techniques in [18,Chapter 3].…”