Liouvillian Solutions of Difference-Differential Equations

Ruyong Feng; Michael F. Singer; Min Wu

arXiv:0810.1574·cs.SC·October 10, 2008·1 cites

Liouvillian Solutions of Difference-Differential Equations

Ruyong Feng, Michael F. Singer, Min Wu

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TL;DR

This paper introduces the concept of liouvillian solutions for linear difference-differential systems over a field with automorphism and derivation, providing a Galois-theoretic characterization and an algorithm for specific cases.

Contribution

It defines liouvillian solutions for difference-differential systems and offers a Galois group criterion along with an algorithm for systems over complex rational functions.

Findings

01

Characterization of liouvillian solutions via Galois groups

02

Algorithm for deciding liouvillian solutions when the system size is prime

03

Application to systems over C(x,t) with specific automorphism and derivation

Abstract

For a field k $w i t hana u t o m or p hi s mσ an d a d er i v a t i o n δ, w e in t r o d u ce t h e n o t i o n o f l i o uv i l l ian so l u t i o n so f l in e a r d i f f er e n ce - d i f f er e n t ia l sy s t e m s σ (Y) = A Y, δ (Y) = B Y o v er k an d c ha r a c t er i z e t h ee x i s t e n ceo f l i o uv i l l ian so l u t i o n s in t er m so f t h e G a l o i s g r o u p o f t h esy s t e m s . W e w i l l g i v e ana l g or i t hm t o d ec i d e w h e t h er s u c ha sy s t e mha s l i o uv i l l ian so l u t i o n s w h e nk = C (x, t), σ (x) = x + 1, δ = d / d t$ and the size of the system is a prime.

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Taxonomy

TopicsPolynomial and algebraic computation · Algebraic Geometry and Number Theory · Advanced Differential Equations and Dynamical Systems