We present a product formula for the initial parts of the sparse resultant associated with an arbitrary family of supports, generalizing a previous result by Sturmfels. This allows to compute the homogeneities and degrees of this sparse resultant, and its evaluation at systems of Laurent polynomials with smaller supports. We obtain an analogous product formula for some of the initial parts of the principal minors of the Sylvester-type square matrix associated with a mixed subdivision of a polytope. Applying these results, we prove that under suitable hypothesis, the sparse resultant can be computed as the quotient of the determinant of such a square matrix by one of its principal minors. This generalizes the classical Macaulay formula for the homogeneous resultant and confirms a conjecture of Canny and Emiris.

我们提出了与任意支持族相关的稀疏结果的初始部分的乘积公式，概括了 Sturmfels 先前的结果。这允许计算这个稀疏结果的同质性和程度，以及它在具有较小支持的 Laurent 多项式系统上的评估。我们获得了与多面体的混合细分相关的 Sylvester 型方阵的主要次要部分的一些初始部分的类似乘积公式。应用这些结果，我们证明在适当的假设下，稀疏结果可以计算为这种方阵的行列式与其主次要之一的商。这推广了齐次结果的经典麦考利公式，并证实了 Canny 和 Emiris 的猜想。