尺度支撑系统的多齐次结式
Multihomogeneous Resultant Formulae for Systems with Scaled Support
摘要：
Weyman、Zelevinsky、Sturmfels、Dickenstein和Emiris研究了非混合系统的多齐次（或多级）结式矩阵的构造方法。我们将这些结构推广到混合系统，其牛顿多面体是一个多面体的缩放副本，从而向具有任意支撑的系统迈出了一步。首先，我们指定矩阵的行列式等于结果和特点的系统，承认这样的公式。Bezout型行列式公式不存在，但我们描述所有可能的西尔维斯特型和混合公式。我们建立严格的界限，所有相应的度向量，并指定域，肯定会包含这样的向量，后者是新的，即使是未混合的情况下。其次，我们利用乘法表和强对偶理论明确地指定结果矩阵，对于一般的标度系统，从而包括非混合系统。遇到的矩阵进行分类，其中包括一种新型的西尔维斯特型矩阵以及Bezout型矩阵，称为部分Bezoutians。我们的公共领域Maple实现包括在内存中有效存储复合体，并构建结果矩阵。
英文摘要：
Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. Bezout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for all corresponding degree vectors, and specify domains that will surely contain such vectors; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, for a general scaled system, thus including unmixed systems. The encountered matrices are classified; these include a new type of Sylvester-type matrix as well as Bezout-type matrices, known as partial Bezoutians. Our public-domain Maple implementation includes efficient storage of complexes in memory, and construction of resultant matrices.
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