Bounds of heights of coefficients of rational polynomials

Question

For a non zero rational $r=p/q$ ($p,q\in\mathbb Z$ coprimes), define the height of $r$ by $\mathrm{ht}(r)=\max(|p|,|q|)$ (by convention $\mathrm{ht}(0)=0$). For a polynomial $P\in\mathbb Q[X]$, define the height of $P$ by the maximum of height of its coefficients. Let $A$ and $C$ be two non zero polynomials of $\mathbb Z[X]$ such that $A$ divides $C$ in $\mathbb Q[X]$. Denote by $B$ the quotient of the division of $C$ by $A$. My question: can one bound the height of $B$ in function of the height of $A$ and $C$?

Thanks in advance for any answer

Answer 1

In general, no. Take $A=(x-1)^2$, $C=(x^n-1)^2$ for large $n$. Then $B=(1+x+\ldots+x^{n-1})^2$ has height $n$.

Answer 2

Several results are given in Panaitopol and Stefanescu, Inequalities on polynomial heights, Journal of Inequalities in Pure and Applied Mathematics, volume 2, issue 1, article 7, 2001, available at https://www.emis.de/journals/JIPAM/images/017_00_JIPAM/017_00_www.pdf

For example, if $\mu<2$ is a lower bound of the moduli of the roots of the complex polynomial $P$, and $Q$ is a proper divisor of $P$, then $H(Q)<(2/\mu)^n\,H(P)$.

The paper also gives references to earlier work by Beauzamy, by Mignotte, and by others.