On the computation of the difference Galois groups of order three equations

2022/08/29 by Thomas Dreyfus, Dreyfus, Thomas, Marina Poulet +1
Computer Science · Mathematics · Physics and Astronomy · #12H05 #39A06 #Commutative Algebra (math.AC) #FOS: Mathematics #Nonlinear Waves and Solitons #Number Theory (math.NT) #Numerical methods for differential equations #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2208.13448

openalex publication_date 2022/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02

Abstract

In this paper we consider the problem of computing the difference Galois groups of order three equations for a large class of difference operators including the shift operator (Case S), the q-difference operator (Case Q), the Mahler operator (Case M) and the elliptic case (Case E). We show that the general problem can be reduced to several ancillary problems. We prove criteria to detect the irreducible and imprimitive Galois groups. Finally, we give a sufficient condition of differential transcendence of solutions of order three difference equations. We also compute the difference Galois group of an equation suggested by Wadim Zudilin.

Citations

• On the algebraic relations between Mahler functions
• The q-analogue of the wild fundamental group (I)

Related

• Analytic q-difference equations
• Hypertranscendance et Groupes de Galois aux differences
• On the Computation of the Galois Group of Linear Difference Equations
• On the Definitions of Difference Galois Groups
• Galois Groups of Linear Difference-Differential Equations