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\section*{6.885 Problem Set: Due November 14, 2005}
\subsection*{Instructions}
\begin{itemize}
\item If you are taking the course for credit, you must turn
in this problem set.
\item You should do this in pairs. If you have trouble finding
a partner let me know and I'll help you find one.
\item While a pair may ask around for specific questions (or
consult the literature), you should try to solve the problem
set on your own, to the maximum extent possible. If you do
consult others, you *must* list all collaborators and/or sources.
\item Coherence and clarity of the writeup are important, so
please factor in at least 50\% of the time to do the writing.
Iterate with your partner on the writing, so that one person
goes over the other's writeup and if possible improves on it.
\item Answers are due before lecture on Monday, November 14.
\end{itemize}
\subsection*{Background: Factoring polynomials over the rationals}
Recall that our goal is to verify various claims needed to
assert the correctness of the following
factoring algorithm for polynomials over the rationals.
Given a polynomial $f \in \Z[X]$ of degree at most d and
coefficients bounded in magnitude by $2^n$, we find a factor
of $f$ as follows:
\begin{enumerate}
\item Let $f'$ denote the derivative of $f$.
If the greatest common divisor of $f$ and $f'$ is nontrivial,
output that factor and stop.
\item Find a prime $p$ such that the gcd of the images of
$f$ and $f'$ in $\F_p[X]$ is also trivial.
\item Factor $f$ as $f = gh \pmod{p}$ in $\F_p[X]$,
where $g$ and $h$ are monic and relatively
prime and $g$ is irreducible in $\F_p[X]$.
\item Use Hensel Lifting to find monic polynomials $G$ and $H$
such that $f = GH \pmod{p^t}$, with $G = g \pmod{p}$, and
$H = h \pmod{p}$.
\item Find $\bar{g}$ and $\bar{G}$ of appropriate degree such that
$\bar{g}(X) = G(X)\bar{G}(X) \pmod{p^t}$.
\item If $\bar{g}$ and $f$ have a nontrivial gcd, output that gcd; otherwise, output
``f is irreducible.''
\end{enumerate}
\subsection*{The actual problems}
\begin{enumerate}
\item {\bf (Resultants of integer polynomials)} Recall that
for relatively prime polynomials $f,g \in \Z[X]$, we defined
the resultant of $f$ and $g$ to be the smallest positive integer $R$
such that $R = af + bg$, where $a, b \in \Z[X]$.
Prove that if $f$ and $g$ are of degree at most $d$, with coefficients
in the range $-C,\ldots,+C$, then their resultant is bounded by
$O(dC)^{O(d)}$.
{\bf Hint:} Proceed as in the proof of the bivariate case, substituting
the pigeonhole principle for one of the dimension counting arguments
used in the bivariate case.
\item {\bf (Coefficients of factors)}
Prove that if $f = gh$ over $\Z[X]$, with the degree of $f$ at most $d$
and the coefficients of $f$ are in the range $-C,\ldots,+C$, then the
coefficients of $g$ are bounded by $O(d)^{O(d)}\cdot C$.
\item {\bf (Hensel lifting)}
Let $f = g^* h^*$, with $g^*$ being irreducible over the
integers. Furthermore, let $g^*$ factor into irreducible
polynomials of the form $g(x)g_1(x)\cdots g_k(x) \pmod{p}$.
Show that $g^*$ is a candidate polynomial
for the $\bar{g}$ of Step 5 of the algorithm.
\item {\bf (Resultants and validity of the solution)}
Prove that the $g^*$ of Exercise 3 divides any
$\bar{g}$ reported in Step 5 of the algorithm.
\item {\bf (Summary)} Use the above exercise to conclude the
correctness of the algorithm above. Among other things, you
must bound the running time of the algorithm by some polynomial
in $n$ and $d$.
\end{enumerate}
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