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Let $P\in {\mathbb Z}[X]$ be a polynomial,

$$ P=\sum_{k=0}^{n} a_kx^k $$ Let us put $$ || P || = \max_{0 \leq k \leq n} |a_k| $$

Let $Q$ be a factor of $P$. Can we bound $||Q||$ by some function of $||P||$ ? If so, is an asymptotically optimal bound known ?

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I assume $Q\in \mathbb Z[X]$ as well? –  Alex Becker Dec 20 '12 at 21:27
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I changed $\displaystyle{\sf max}_{0\le k\le n}$ to $\displaystyle\max_{0\le k\le n}$. That is standard usage, and what you actually type is shorter and simpler: just \max_{0\le k\le n} instead of {\sf max}_{0\le k\le n}, and in a "displayed", as opposed to "inline" setting, it puts the subscript directly below $\max$. –  Michael Hardy Dec 20 '12 at 22:56
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1 Answer

up vote 6 down vote accepted

No.

Take for instance

$Q=x+2x^2+3x^3+4x^4+...+(n-1)x^{n-1}+nx^n+(n-1)x^{n+1}+...+2x^{n-2}+x^{2n-1}$,

for some $n$.

Then let $P=Q(x-1)^2=x^{2n+1}-2x^{n+1}+x$.

In this case, $||P||=2$ but $||Q||$ can be made as large as we want.

Note: it has been proven that cyclotomic polynomials have arbitrarily large coefficients. Cyclotomic polynomials are all factors of $x^n-1$ for some $n$. Therefore, even if $P$ is in the very simple form $x^n-1$, and $||P||=1$, there is no bound on the coefficients of factors of $P$.

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