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.

翻译：提出了一种新算法，用于计算与多项式 $a$ 和 $b$（在 $\mathbb F_q[x,y]$ 中，且无非常数公因子）关联的西尔维斯特矩阵（关于 $x$ 或 $y$）的最大次数不变因子。该算法为蒙特卡洛型随机算法，所需比特运算量为 $O((de)^{1+\epsilon}\log(q) ^{1+o(1)})$，其中 $d$ 和 $e$ 分别限制输入在 $x$ 和 $y$ 方向上的次数。由此可得，以下计算具有相同的复杂度估计：当多项式系统 $a=b=0$ 在无穷远处无根时，生成消去理想 $\langle a,b \rangle \cap \mathbb F_q[x]$（或 $\mathbb F_q[y]$）的生成元；当 $a$ 和 $b$ 足够一般（特别地，西尔维斯特矩阵具有唯一的非平凡不变因子）时，计算 $a$ 和 $b$ 的结式。我们的方法是将问题约化为商代数 $\mathbb F_q[x,y]/\langle a,b \rangle$ 中的最小多项式问题。通过提出一种基于结构化多项式矩阵除法的新方法来处理商代数中的元素计算，我们成功改进了当前已知的最佳复杂度界。