Statement

Let $f$ and $g$ in ${\mbox{${\mathbb{Z}}$}}[x]$ be two polynomials with respective degrees $m \geq n > 0$ . We assume that the leading coefficient ${\rm lc}(g)$ of $g$ divides the leading coefficient ${\rm lc}(f)$ of $f$ . We want to decide whether $g$ divides $f$ exactly, that is, whether there exists a polynomial $q \in {\mbox{${\mathbb{Z}}$}}[x]$ such that $f = q g$ holds. Moreover, if $g$ divides $f$ exactly, we want to compute the quotient $q$ .

Another obvious necessary condition (for $g$ to divide $f$ exactly) is that the trailing coefficient of $g$ (that is the coefficient of $g$ of its non-zero term with smallest degree) divides that of $f$ . So, we assume that this condition holds too.

If $f$ and $g$ are random enough the probability for $g$ to divide $f$ exactly is clearly low. So we aim at designing a modular method that will take advantage of this remark.