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A new algorithm is presented for computing the largest degree invariantfactor of the Sylvester matrix (with respect either to $x$ or $y$) associatedto two polynomials $a$ and $b$ in $\mathbb F_q[x,y]$ which have no non-trivialcommon divisors. The algorithm is randomized of the Monte Carlo type andrequires $O((de)^{1+\epsilon}\log(q) ^{1+o(1)})$ bit operations, where $d$ an$e$ respectively bound the input degrees in $x$ and in $y$. It follows that thesame complexity estimate is valid for computing: a generator of the eliminationideal $\langle a,b \rangle \cap \mathbb F_q[x]$ (or $\mathbb F_q[y]$), as soonas 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 theSylvester matrix has a unique non-trivial invariant factor. Our approach is touse the reduction of the problem to a problem of minimal polynomial in thequotient algebra $\mathbb F_q[x,y]/\langle a,b \rangle$. By proposing a newmethod based on structured polynomial matrix division for computing with theelements in the quotient, we manage to improve the best known complexitybounds.

Elimination ideal and bivariate resultant over finite fields