Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula when we can express it as a determinant of a matrix whose elements are the coefficients of the input polynomials. We study the resultant in the context of mixed multilinear polynomial systems, that is multilinear systems with polynomials having different supports, on which determinantal formulas were not known. We construct determinantal formulas for two kind of multilinear systems related to the Multiparameter Eigenvalue Problem (MEP): first, when the polynomials agree in all but one block of variables; second, when the polynomials are bilinear with different supports, related to a bipartite graph. We use the Weyman complex to construct Koszul-type determinantal formulas that generalize Sylvester-type formulas. We can use the matrices associated to these formulas to solve square systems without computing the resultant. The combination of the resultant matrices with the eigenvalue and eigenvector criterion for polynomial systems leads to a new approach for solving MEP.

中文翻译：

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混合多线性系统族的 Koszul 型行列式公式

结果的有效计算是消元理论和多项式系统求解的核心问题。通常，我们将结果计算为矩阵行列式的商，并且当我们可以将其表示为矩阵的行列式时，我们说存在行列式公式，矩阵的元素是输入多项式的系数。我们在混合多重线性多项式系统的背景下研究结果，即多项式具有不同支持的多重线性系统，其行列式公式未知。我们为与多参数特征值问题 (MEP) 相关的两种多线性系统构建行列式：第一，当多项式在除一个变量块之外的所有变量中都一致时；第二，当多项式是具有不同支持度的双线性时，与二部图相关。我们使用 Weyman 复数来构造 Koszul 型行列式公式，以推广 Sylvester 型公式。我们可以使用与这些公式相关的矩阵来求解平方系统，而无需计算结果。将所得矩阵与多项式系统的特征值和特征向量准则相结合，形成了求解 MEP 的新方法。