In this paper we study the equations of the elimination ideal associated with $n+1$ generic multihomogeneous polynomials defined over a product of projective spaces of dimension $n$. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods.

中文翻译：

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多级Sylvester形式，对偶和消除矩阵

在本文中，我们研究与在尺寸为$ n $的射影空间乘积上定义的$ n + 1 $通用多齐次多项式相关的消除理想方程。我们首先证明对偶性，然后通过引入多级Sylvester形式使对偶性明确。这些结果提供了相似性质的部分概括，这在单个投影空间上定义的齐次多项式系统的设置中是已知的。重要的结果是，我们推导出了一个新的消除矩阵族，该族可用于通过线性代数方法求解零维多射影多项式系统。