The zero multiplicity of linear recurrence sequences

Wri te k p(z) = II(z0 (1.2) i=1 with d i s t inc t roo ts c~1, ..., ak . The sequence is said to be nondegenerate if no quot ien t ~i/c~j ( l ~ i < j < ~ k ) is a root of 1. The zero multiplicity of the sequence is the number of n E Z wi th Un=O. For an i n t roduc t ion to l inear recurrences and exponen t i a l equat ions , see [10]. A classical t heo rem of Skolem Mahle r Lech [4] says t h a t a nondegene ra t e l inear recurrence sequence has finite zero mul t ip l ic i ty . Schlickewei [6] and van der Poo r t e n and Schlickewei [5] der ived uppe r bounds for the zero mul t ip l i c i ty when the me mbe r s of the sequence lie in a number field K . These bounds d e p e n d e d on the order t, the degree of K , as well as on the number of d i s t inc t p r ime ideal factors in the decompos i t i on of the f rac t ional ideals (c~) in K . More recently, Schlickewei [7] gave bounds which depend