Hints

• For both Exercises 1 and 2, looking at Exercise 5.34 from [GG99] should be helpful.
• For computing the bound $B_q$ , looking at Exercise 2 of Quiz 1 of last year can be helpful too. This bound should look like $B_f B_g^{m -n + 1}$ .
• For Exercise 1, it is important to remember the following fact. Let $a$ and $b$ be two polynomials in ${\mbox{${\mathbb{Z}}$}}[x]$ or ${\mbox{${\mathbb{Z}}$}}/m{\mbox{${\mathbb{Z}}$}}[x]$ (for some integer $m > 1$ , prime or not) such that the leading coefficient of $b$ is one. Then, there exists a unique couple $(q, r)$ such that $a = b q + r$ and ${\deg}(r) < {\deg}(b)$ . The algorithm for the division is in the first section of the Chapter From Newton to Hensel on our web site.