Elimination ideal and bivariate resultant over finite fields

Abstract

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.