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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?

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    If you want to be sure I see your comment, you have to put @Gerry in there somewhere. Anyway, yes, that's a linear homogeneous constant-coefficient recurrence. I trust that by now you've had a look at Skolem-Mahler-Lech. Has it been helpful?2012-09-12

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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.