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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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There is no way to answer such a vague question. What kind of recurrence relation do you have? – Marc van Leeuwen Sep 11 '12 at 12:41
If it's a linear homogeneous constant-coefficient recurrence, the Skolem-Mahler-Lech Theorem is helpful. See, e.g., – Gerry Myerson Sep 11 '12 at 12:52
thanks. I mean given a linear recurrence relation, e.g., $a_{n+m}= \sum_{i=1}^{m-1} c_ia_{n+i}$, and $c_i$ are all integers, and $a_1, \dots, a_{m-1}$ are given integers. – user29271 Sep 11 '12 at 15:39
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? – Gerry Myerson Sep 12 '12 at 5:58

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.

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