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Hensel's Lemma gives us a sequence $(a_n)$ of solutions of $f(a_n) \equiv 0 \pmod {p^n}$, so in particular $p^n|f(a_n)$. Is there anything known about the sequence $o_n = \frac{f(a_n)}{p^n}$?

It seemed to be related to this

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I just want to say that if the root is irrational then the sequence of digits of the p-adic number is not periodic... so $o_n$ must be unbounded (when considered as an integer, it would be bounded as a p-adic number). In the rational case $o_n$ must be periodic and bounded. Thanks to @anon. –  sperners lemma Nov 2 '12 at 16:28
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I don't think this makes sense, because $a_n$ is only defined modulo $p^n$, and hence the same is true of $f(a_n)$.

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Good point. I think you can iteratively apply the transformation $x\mapsto x-f(x)/f'(x)$, i.e. use Newton's method to create a sequence of successive approximations to a $p$-adic root to $f(x)$, as long as you pick an initial seed value in $\Bbb Z_p$, though. In which case discussions about $m_n=v_p(f(a_n))$ and $p^{-m_n}f(a_n)$ are sensible in the $p$-adics. –  anon Nov 2 '12 at 20:19
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