# Getting a bound on the coefficients of the factor polynomial

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 $i=0,\ldots ,m$ ?

-
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

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