We exploit structure in polynomial system solving by considering polynomials that are linear in subsets of the variables. We focus on algorithms and their Boolean complexity for computing isolating hyperboxes for all the isolated complex roots of well-constrained, unmixed systems of multilinear polynomials based on resultant methods. We enumerate all expressions of the multihomogeneous (or multigraded) resultant of such systems as a determinant of Sylvester-like matrices, aka generalized Sylvester matrices. We construct these matrices by means of Weyman homological complexes, which generalize the Cayley-Koszul complex.

The computation of the determinant of the resultant matrix is the bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector multiplication, which corresponds to multivariate polynomial multiplication, by extending the seminal work on Macaulay matrices of Canny, Kaltofen, and Yagati Canny et al. (1989) to the multihomogeneous case.

We compute a rational univariate representation of the roots, based on the primitive element method. In the case of 0-dimensional systems we present a Monte Carlo algorithm with probability of success $1-1/2^{\varrho }$, for a given $\varrho \ge 1$, and bit complexity ${\tilde{\mathcal{O}}}_B(n^2{D}^{4+ϵ}(n^{N+1}+\tau )+n{D}^{2+ϵ}\varrho (D+\varrho ))$ for any $ϵ>0$, where n is the number of variables, D equals the multilinear Bézout bound, N is the number of variable subsets, and τ is the maximum coefficient bitsize. We present an algorithmic variant to compute the isolated roots of overdetermined and positive-dimensional systems. Thus our algorithms and complexity analysis apply in general with no assumptions on the input.

我们通过考虑在变量子集中线性的多项式来利用多项式系统求解中的结构。我们关注于算法和它们的布尔复杂度，用于基于结果方法为约束良好的，未混合的多元线性多项式系统的所有隔离的复杂根计算隔离超框。我们列举了这类系统的多均质（或多级）结果的所有表达式，作为类Sylvester矩阵（也称为广义Sylvester矩阵）的行列式。我们通过韦曼同源复合物来构造这些矩阵，这些复合物将Cayley-Koszul复杂化。

所得矩阵行列式的计算是整体复杂性的瓶颈。通过扩展对Canny，Kaltofen和Yagati Canny等人的Macaulay矩阵的开创性工作，我们利用准Toeplitz结构将问题简化为有效的矩阵矢量乘法，该乘法对应于多元多项式乘法。（1989）的多同质情况。

我们基于原始元素方法计算根的有理单变量表示形式。在零维系统的情况下，我们提出具有成功概率的蒙特卡洛算法$\mathrm{1个}-\mathrm{1个}/2^{\varrho }$，对于给定 $\varrho \ge \mathrm{1个}$和位复杂度 ${\stackrel{〜}{\mathcal{Ø}}}_{乙}（{ñ}^2{d}^{4+ϵ}（{ñ}^{ñ+\mathrm{1个}}+\tau ）+ñd^{2+ϵ}\varrho （d+\varrho ））$ 对于任何 $ϵ>0$，其中n是变量的数量，D等于多线性Bézout边界，N是变量子集的数量，并且τ是最大系数位大小。我们提出了一种算法变体来计算超定和正维系统的孤立根。因此，我们的算法和复杂性分析通常在不假设输入的情况下适用。