Primitive divisors of Lucas sequences in polynomial rings

Abstract

It is known that all terms $U_n$ of a classical regular Lucas sequence have a primitive prime divisor if $n>30$ [2]. In addition, a complete description of all regular Lucas sequences and their terms $U_n$, $2\le n\le 30$, which do not have a primitive divisor is also known. Here, we prove comparable results for Lucas sequences in polynomial rings, correcting some previous theorem on the same subject. The first part of our paper develops some elements of Lucas theory in several abstract settings before proving our main theorem in polynomial rings.