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
Best Answer
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 $n$.