zeros of linear recurence sequences Given a linear recurrence sequence $\{a_n\}_{n\geq 0}$, how to decide whethere there are infinitely many zeros, or there are only finitely many ones?

The characteristic polynomial of your recurrence is $$x^{m-1}-c_{m-1}x^{m-2}-\cdots-c_1$$
Here's something that follows from the Skolem-Mahler-Lech Theorem: if the recurrence has infinitely many zeros, then the characteristic polynomial has two distinct roots whose ratio is a root of unity. 
Careful: this is not an "if and only if". 
Another good source is Chapter 2 of the book Recurrence Sequences by Graham Everest, Alf Van Der Poorten, Igor Shparlinski and Thomas Ward.
