Galois Theory of Difference Equations with Periodic Parameters

σ-Galois theory of linear difference equations

Inspired by the numerous applications of the differential algebraic independence results from [36], 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 ∈ GLn(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 ...

Galois theory of fuchsian q-difference equations

We propose an analytical approach to the Galois theory of singular regular linear q-difference systems. We use Tannaka duality along with Birkhoff’s classification scheme with the connection matrix to define and describe their Galois groups. Then we describe fundamental subgroups that give rise to a Riemann-Hilbert correspondence and to a density theorem of Schlesinger’s type.

Nonlinear Difference Equations with Periodic Solutions

For ten nonlinear difference equations with only p-periodic solutions it is shown that the characteristic polynomials of the corresponding linearized equations about the equilibria have only zeros which are p-th roots of unity. An analogous result is shown concerning two systems of such equations. Five counterexamples show that the reverse is not true. Some remarks are made concerning equations...

Galois Theory and Painlevé Equations

— The paper consists of two parts. In the first part, we explain an excellent idea, due to mathematicians of the 19-th century, of naturally developing classical Galois theory of algebraic equations to an infinite dimensional Galois theory of nonlinear differential equations. We show with an instructive example how we can realize the idea of the 19-th century in a rigorous framework. In the sec...

Galois Theory of Linear Differential Equations