Difference Galois theory of linear differential equations

Vizio, Lucia Di; Hardouin, Charlotte; Wibmer, Michael

doi:10.1016/j.aim.2014.04.005

Cited by 26 publications

(74 citation statements)

References 36 publications

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“…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.…”

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confidence: 99%

-Galois Theory of Linear Difference Equations

Ovchinnikov¹,

Wibmer²

2014

International Mathematics Research Notices

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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...

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confidence: 99%

-Galois Theory of Linear Difference Equations

Ovchinnikov¹,

Wibmer²

2014

International Mathematics Research Notices

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“…. , D k−1 ∈ GL n KF k−1 that satisfy (13), (14), and (15) (k − 1 is substituted for r). We claim that there exists a Σ-field F k generated over K Φ,∆ by at most (k − 1)n 2 elements and Z ∈ GL n (KF k ) such that…”

Section: Resultsmentioning

confidence: 99%

“…satisfies (13), (14), and (15) (k is substituted for r). We need to construct a Σ-field F k and Z ∈ GL n (KF k ) such that…”

Section: Resultsmentioning

confidence: 99%

Difference integrability conditions for parameterized linear difference and differential equations

Ovchinnikov

2014

Advances in Applied Mathematics

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This paper is devoted to integrability conditions for systems of linear difference and differential equations with difference parameters. It is shown that such a system is difference isomonodromic if and only if it is difference isomonodromic with respect to each parameter separately. Due to this result, it is no longer necessary to solve non-linear difference equations to verify isomonodromicity, which will improve efficiency of computation with these systems.

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“…Difference algebraic groups occur as Galois groups of linear differential or difference equations depending on a discrete parameter ( [DVHW14], [OW15]). We expect that, via these Galois theories, a better understanding of the components of difference algebraic groups will contribute to a better understanding of the difference algebraic relations among the solutions of linear differential or difference equations.…”

Section: Introductionmentioning

confidence: 99%