-Galois Theory of Linear Difference Equations

Abstract

We develop a Galois theory for systems of linear difference equations with an action of an endomorphism σ. This provides a technique to test whether solutions of such systems satisfy σ-polynomial equations and, if yes, then characterize those. We also show how to apply our work to study isomonodromic difference equations and difference algebraic properties of meromorphic functions. IntroductionInspired by the numerous applications of the differential algebraic independence results from [34], we develop a Galois theory with an action of an endomorphism σ for systems of linear difference equations of the form φ(y) = Ay, where A ∈ GL n (K) and K is a φσ-field, that is, a field with two given commuting endomorphisms φ and σ, like in Example 2.1. This provides a technique to test whether solutions of such systems satisfy σ-polynomial equations and, if yes, then characterize those. Galois groups in this approach are groups of invertible matrices defined by σ-polynomial equations with coefficients in the σ-field K φ := {a ∈ K | φ(a) = a}. In more technical terms, such groups are functors from K φσ-algebras to sets represented by finitely σ-generated K φ -σ-Hopf algebras [22] . Also, our work is a highly non-trivial generalization of [5], where similar problems were considered but σ was required to be of finite order (there exists n such that σ n = id).Our main result is a construction of a σ-Picard-Vessiot (σ-PV) extension (see Theorem 2.14), that is, a minimal φσ-extension of the base φσ-field K containing solutions of φ(y) = Ay. It turns out that the standard constructions and proofs in the previously existing difference Galois theories do not work in our setting. Indeed, this is mainly due to the reason that even if the field K φ is σ-closed [52], consistent systems of σ-equations (such that the equation 1 = 0 is not a σ-algebraic consequence of the system) with coefficients in K φ might not have a solution with coordinates in K φ (see more details in Remarks 2.10 and 2.12). However, our method avoids this issue. In our approach, a σ-PV extension is built iteratively (applying σ), by carefully choosing a suitable usual PV extension [46] at each step, and then "patching" them together. This is a difficult problem and requires several preparatory steps as described in §2.4. A similar approach was also taken in [57, Thm. 8] for systems of differential equations with parameters. However, our case is more subtle and, as a result, requires more work.Galois theory of difference equations φ(y) = Ay without the action of σ was studied in [46,10,1,2,3,4,58], with a non-linear generalization considered in [30,41], as well as with an action of a derivation ∂ in [31,32,34,18,20,19,21,17,16]. The latter works provide algebraic methods to test whether solutions of difference equations satisfy polynomial differential equations (see also [38] for a general Tannakian approach). In particular, these methods can be used to prove Hölder's theorem that states that the Γ-function, which satisfies the difference equation Γ(x+ 1) = x·Γ(x), s...

Consistent systems of linear differential and difference equations

“…Thus we have shown that (33) is equivalent over C(x) to σ(Z) = A 0 Z with some constant invertible matrix A 0 . To complete the proof of Theorem 13 for case 2Q, apply the same gauge transformation to σ 2 (Y ) = B 2 (x)Y to obtain σ 2 (Z) =B 2 (x)Z.…”

On the computation of the parameterized differential Galois group for a second-order linear differential equation with differential parameters

“…In the applications of the PPV theory, it is often useful to know whether any of the solutions to (12) are Π-transcendental (see e.g. Arreche, 2013;Hardouin, 2008;Hardouin and Singer, 2008;Ovchinnikov and Wibmer, 2015). We denote by D := F •Π the Lie vector space of parametric derivations obtained by taking the F -linear span of Π.…”

Action of an endomorphism on (the solutions of) a linear differential equation

“…The dependence of a full set of solutions of a linear functional equation with respect to a discrete parameter is considered in [8], [9] and [18]. Due to the intrinsic difficulty of difference algebra, the proofs, but also sometimes the statements, are more complicated than in the analogous continuous theory.…”