  A new algorithm is presented for computing the largest degree invariant
factor of the Sylvester matrix (with respect either to $x$ or $y$) associated
to two polynomials $a$ and $b$ in $\mathbb F_q[x,y]$ which have no non-trivial
common divisors. The algorithm is randomized of the Monte Carlo type and
requires $O((de)^{1+\epsilon}\log(q) ^{1+o(1)})$ bit operations, where $d$ an
$e$ respectively bound the input degrees in $x$ and in $y$. It follows that the
same complexity estimate is valid for computing: a generator of the elimination
ideal $\langle a,b \rangle \cap \mathbb F_q[x]$ (or $\mathbb F_q[y]$), as soon
as the polynomial system $a=b=0$ has not roots at infinity; the resultant of
$a$ and $b$ when they are sufficiently generic, especially so that the
Sylvester matrix has a unique non-trivial invariant factor. Our approach is to
use the reduction of the problem to a problem of minimal polynomial in the
quotient algebra $\mathbb F_q[x,y]/\langle a,b \rangle$. By proposing a new
method based on structured polynomial matrix division for computing with the
elements in the quotient, we manage to improve the best known complexity
bounds.
