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.

Marc Moreno Maza