论文标题
感谢您的Sylvester形式
Toric Sylvester forms
论文作者
论文摘要
在本文中，我们研究了稀疏的均质多项式在投影式复曲面$ x $上产生的理想饱和的结构。作为我们的主要结果，我们在$ x $上的一定阳性假设下建立了双重性属性，并通过引入复曲面的Sylvester表格来明确。特别是，我们证明了折叠的sylvester形式的某些分级组成部分的产量基库，$ i^{\ text {sat}}/i $，其中$ i $表示由$ n+1 $ n $形式产生的理想，$ n $是$ x $和$ x $ and $ i^{s sat} $ irree a $ irree的$ x $的维度$ x $。然后，为了说明感谢您的光滑折叠品种消除理论的三种后果：（1）我们引入了一个新的消除矩阵家族，可用于通过线性代数方法，包括过度确定的多项式多项式系统来求解稀疏的多项式系统； （2）通过将复曲面的sylvester形式纳入与多项式系统相关的经典koszul复合物中，我们获得了稀疏结果的新表达式作为复合物的决定因素； （3）我们为计算两种形式的乘积的感谢您的曲折残基发出了新的公式。
In this paper, we investigate the structure of the saturation of ideals generated by sparse homogeneous polynomials over a projective toric variety $X$ with respect to the irrelevant ideal of $X$. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on $X$. In particular, we prove that toric Sylvester forms yield bases of some graded components of $I^{\text{sat}}/I$, where $I$ denotes an ideal generated by $n+1$ generic forms, $n$ is the dimension of $X$ and $I^{\text{sat}}$ the saturation of $I$ with respect to the irrelevant ideal of the Cox ring of $X$. Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory over smooth toric varieties: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we explote a new formula for computing toric residues of the product of two forms.
