论文标题
Hadamard产品和二项式理想
Hadamard Products and Binomial Ideals
论文作者
论文摘要
我们研究了两种品种$ v $和$ w $的Hadamard产品，特别关注$ v $和$ w $中的一个或两者是二项二元品种时的情况。本文的主要结果表明，当$ v $和$ w $都是二项式品种时，定义$ v $和$ w $的二项式具有相同的二项式指数，则可以明确地计算出$ v \ star w $的定义方程式，并直接从$ v $和$ v $和$ w $的定义方程式中直接计算。该结果恢复了有关Hadamard二项式超曲面和复曲面品种的已知结果。此外，作为我们的主要结果的应用，我们描述了图$ g $的感谢您的hadamard产品与$ g $ g $的$ g $的福利$ i_h $之间的关系。我们还得出了有关Hadamard产品代数不变的结果：假设$ v $和$ w $具有相同的指数，我们表明$ \ text {deg}（v \ star w）= \ text {deg}（deg}（v）= \ text {deg}（deg}（w}（w）$ and $ \ dim（v）最后，给定任何（不一定是二项式的）投影变化$ v $和一个点$ p \ in \ mathbb {p}^n \ setMinus \ setMinus \ mathbb {v}（x_0x_1 \ cdots x_n）$ V $。
We study the Hadamard product of two varieties $V$ and $W$, with particular attention to the situation when one or both of $V$ and $W$ is a binomial variety. The main result of this paper shows that when $V$ and $W$ are both binomial varieties, and the binomials that define $V$ and $W$ have the same binomial exponents, then the defining equations of $V \star W$ can be computed explicitly and directly from the defining equations of $V$ and $W$. This result recovers known results about Hadamard products of binomial hypersurfaces and toric varieties. Moreover, as an application of our main result, we describe a relationship between the Hadamard product of the toric ideal $I_G$ of a graph $G$ and the toric ideal $I_H$ of a subgraph $H$ of $G$. We also derive results about algebraic invariants of Hadamard products: assuming $V$ and $W$ are binomial with the same exponents, we show that $\text{deg}(V\star W) = \text{deg}(V)=\text{deg}(W)$ and $\dim(V\star W) = \dim(V)=\dim(W)$. Finally, given any (not necessarily binomial) projective variety $V$ and a point $p \in \mathbb{P}^n \setminus \mathbb{V}(x_0x_1\cdots x_n)$, subject to some additional minor hypotheses, we find an explicit binomial variety that describes all the points $q$ that satisfy $p \star V = q\star V$.
