Galois Theory of Difference Equations with Periodic Parameters

Abstract

We develop a Galois theory for systems of linear difference equations with periodic parameters, for which we also introduce linear difference algebraic groups. We apply this to constructively test if solutions of linear q-difference equations, with q ∈ ℂ* and q not a root of unity, satisfy any polynomial ζ-difference equations with ζ ^t  = 1, t ≥ 1.

Key Words:

• Difference Galois theory
• q-Difference equations
• Difference pseudofields

2010 Mathematics Subject Classification:

• Primary 12H10
• Secondary 13N99, 20H25, 39A13

ACKNOWLEDGMENTS

We are grateful to Yves André, Henri Gillet, Sergey Gorchinskiy, Charlotte Hardouin, Manuel Kauers, Alexander Levin, Alice Medvedev, Eric Rosen, Jacques Sauloy, Michael Singer, Lucia di Vizio, Michael Wibmer, and the referee for their helpful comments.

Notes

¹Our proofs apply to the nonseparable case by using nonreduced Hopf algebras, as pointed out to us by Michael Wibmer.

²We also assume this in Section 3, but it is only needed to guarantee the uniqueness of a parameterized PV-extension, which we do not use in the applications. It also implies the existence, but is only a sufficient condition, and there are situations when one does not have to make this assumption. The Galois correspondence in its Hopf-algebraic version does not need this as we show below.

Communicated by K. Misra.