Liouvillian Solutions of Difference-Differential Equations

2008/10/09 by Ruyong Feng, Feng, Ruyong, Michael F. Singer +3
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #FOS: Computer and information sciences #FOS: Mathematics #Polynomial and algebraic computation #Symbolic Computation (cs.SC) #cs.SC #math.CA

paper · pdf · doi:10.48550/arxiv.0810.1574

53 pages

arxiv created 2008/10/09 · openalex publication_date 2008/10/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a field kwith an automorphism σand a derivation δ, we introduce the notion of liouvillian solutions of linear difference-differential systems σ(Y) = AY, δ(Y) = BY over k and characterize the existence of liouvillian solutions in terms of the Galois group of the systems. We will give an algorithm to decide whether such a system has liouvillian solutions when k = C(x,t), σ(x) = x+1, δ= d/dt and the size of the system is a prime.

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• Liouvillian Solutions of Third Order Differential Equations
• Galois Groups of Linear Difference-Differential Equations
• Solvability of linear differential systems in the Liouvillian sense
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