2016/07/29 by Carlos E. Arreche, Michael F. Singer
Computer Science · Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebra over a field #Galois group #Integrable system #Mathematics #Nonlinear Waves and Solitons #Polynomial and algebraic computation #Pure mathematics #math.AC #math.CA #math.RT #msc:12H05 #msc:12H10 #msc:20H20 #msc:39A06 #msc:39A10
paper · pdf · doi:10.1016/j.jalgebra.2017.02.032
published as Journal of Algebra, 480:423-449, (2017)
arxiv created 2016/07/29 · openalex publication_date 2017/03/21 · arxiv updated 2017/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We consider first-order linear difference systems over ℂ(x), with respect to a difference operator σ that is either a shift σ:x↦ x+1, q-dilation σ:x↦ qx with q∈ℂ^× not a root of unity, or Mahler operator σ:x↦ xq with q∈ℤ≥ 2. Such a system is integrable if its solutions also satisfy a linear differential system; it is projectively integrable if it becomes integrable "after moding out by scalars." We apply recent results of Schäfke and Singer to characterize which groups can occur as Galois groups of integrable or projectively integrable linear difference systems. In particular, such groups must be solvable. Finally, we give hypertranscendence criteria.