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Suppose $f(x):=a_0+a_1x+\cdots+a_nx^n$ is a polynomial in $\mathbb{Z}[x]$ and $|a_i|\leq M$ for each $i=0,\ldots ,n$. Now suppose $g(x)$ is a factor of $f(x)$ in $\mathbb{Z}[x]$, then is it possible to get a bound on the coefficients of $g(x)$ in terms of $M$ i.e. if $g(x)=\sum_{i=0}^mb_ix^i$ then does there exist some $M^\prime $, which depends only on $M$, such that $|b_i|\leq M^\prime$ for all $i=0,\ldots ,m$ ?

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Are $n$ and $m$ fixed, or you allow dependence on these parameters? – Davide Giraudo Oct 3 '12 at 15:38
It is quite clear that M' =< M. – mick Oct 3 '12 at 15:40
Yes $M^\prime$ may depend on $n$ and $m$ – pritam Oct 3 '12 at 15:42
If you take x = 1 and factor , what can you conclude ? – mick Oct 3 '12 at 15:42
@mick: taking $x=1$, absolute value of the sum of the coefficients of $g$ is less than the absolute value of the sum of the coefficients of $f$ which is less than $nM$, then ? – pritam Oct 3 '12 at 15:50

Here is one (but it depends on $M$ and $n$):

$||g(x)||_{\infty} \leq 2^n*\sqrt{n+1}*||f(x)||_{\infty}$

This bound (which is very huge) is often used for factoring polynomials using Hensel lifting. It is called the Mignotte-Bound.
See "Maurice Mignotte. Mathematics for Computer Algebra. Springer- Verlag, New York, 1991" for the discussion of this bound.

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