Liouvillian Solutions of Difference-Differential Equations

Ruyong Feng; Michael F. Singer; Min Wu

paper

Liouvillian Solutions of Difference-Differential Equations

arXiv:0810.1574

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.

53 pages