If $p \in \mathbb Z[x]$ has non-negative coefficients $\le n$ and if $q$ is a proper divisor of $p$, are the absolute values of the (integer) coefficients of $q$ bounded by some function of $n$; if so, what is a good bound for the case $n=1$?
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Cyclotomic polynomials divide $p=1+x+\cdots+x^m$ but the (absolute values of) coefficients of cyclotomic polynomials grow unboundedly: see e.g. “ON THE SIZE OF THE COEFFICIENTS OF THE CYCLOTOMIC POLYNOMIAL” by Bateman, available online at https://www.jstor.org/stable/44165422
So there is no bound even in the case $n=1$.
answered Feb 9, 2023 at 18:20
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$\begingroup$ Perhaps the question should ask for bounds that are a function of $n$ and the degree of $p$. $\endgroup$ Commented Feb 9, 2023 at 18:41
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$\begingroup$ @RobertIsrael: well evidently if we fix $n$ and the degree $m$ of $p$ then there will be some bound since there are only finitely many possible $q$. I guess the question is: what kind of explicit bound can you write down. $\endgroup$ Commented Feb 9, 2023 at 18:45
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1$\begingroup$ Thank you for your quick answer to my question. I have an extra condition on $q$ in the case $n=1$, namely that $|q(2)|=1$ which probably rules out the cyclotomic counterexamples you cited. But I have not been able to exploit that yet $\endgroup$ Commented Feb 9, 2023 at 19:45
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1$\begingroup$ Abbott, [Bounds on Factors in Z[x]](arxiv.org/abs/0904.3057) doesn't explicitly address the case of non-negative coefficients but has a lot to say about related questions. $\endgroup$ Commented Feb 10, 2023 at 10:06