On the definitions of difference Galois groups

We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hölder's theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric functions. * IWR, Im Neuenheimer Feld 368,

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“…(cf., [14], Proposition 2.7) If K is a total Σ∆Π-PV ring for equations (5) over k, then Lemma 6.7 and Corollary 6.15 imply the conclusion.…”

Section: σ∆π-Picard-vessiot Extensionsmentioning

confidence: 82%

Differential Galois theory of linear difference equations

Hardouin¹,

2008

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“…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).…”

mentioning

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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“…Then there exists a finite cover of V ′′ of the form V ′ L , where V ′ is a finite cover of V. This is Lemma 4.2 of [9]. The same proof applies in the twisted-periodic case; compare 1.11(3) of Part II.…”

Section: Introductionmentioning

confidence: 72%

Difference fields and descent in algebraic dynamics. II

Chatzidakis¹,

Hrushovski²

2008

J. Inst. Math. Jussieu

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We draw a connection between the model-theoretic notions of modularity (or onebasedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In particular we prove in any dimension a strong dynamical version of Northcott's theorem for function fields, answering a question of Szpiro and Tucker and generalizing a theorem of Baker's for the projective line.The paper comes in three parts. This first part contains an exposition some of the main results of the model theory of difference fields, and their immediate connection to questions of descent in algebraic dynamics. We present the model-theoretic notion of internality in a context that does not require a universal domain with quantifier-elimination. We also note a version of canonical heights that applies well beyond polarized algebraic dynamics. Part II sharpens the structure theory to arbitrary base fields and constructible maps where in part I we emphasize finite base change and correspondences. Part III will include precise structure theorems related to the Galois theory considered here, and will enable a sharpening of the descent results for non-modular dynamics.

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On the definitions of difference Galois groups

Cited by 24 publications

References 26 publications

Differential Galois theory of linear difference equations

Differential Galois theory of linear difference equations

-Galois Theory of Linear Difference Equations

Difference fields and descent in algebraic dynamics. II

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