arXiv math/05086282005-08-31math.ACmath.CO

一种矢量构形的电路理想

The Circuit Ideal of a Vector Configuration

Tristram Bogart, Anders N. Jensen, Rekha R. Thomas

备注：: 25 pages
发表信息：: J. Algebra 309 (2007), no. 2, 518--542
DOI：: 10.1016/j.jalgebra.2006.07.025

摘要：

当$\cc = \cc^+ - \cc^- \in \Z^n$在$\A$的电路上变化时，配置$\A = \{\a_1, ..., \a_n\} \subset \Z^d$的电路理想$\ica$是由二项式${\x}^{\cc^+} - {\x}^{\cc^-} \in \k[x_1, ..., x_n]$生成的理想。这个理想包含在$\A$的复曲面理想$\ia$中，它有许多应用并且计算起来很重要。由于电路可以用线性代数计算，而且这两个理想经常重合，因此有必要了解等式何时出现。在本文中，我们研究$\ica$的关系，$\ia$从各种代数和组合的角度。证明了理想相等的障碍是某些多面体的存在。这一结果是基于$\ica$的单项初始理想的标准对/相关素数的完整表征及其与相应的复曲面初始理想的差异。Eisenbud和Sturmfels证明了$\ia$是$\ica$的唯一极小素数，并且$\ica$的嵌入素数由$\A$所张锥的某些面所索引.我们提供了一个必要条件，为一个特定的面，以指数嵌入素数和部分逆。最后，我们比较了与$\ia$和$\ica$相关的各种多面体风扇。Gröbner风扇的$\ica$示出细化时，$\ia$的余维的理想是最多为2。

英文摘要：

The circuit ideal, $\ica$, of a configuration $\A = \{\a_1, ..., \a_n\} \subset \Z^d$ is the ideal generated by the binomials ${\x}^{\cc^+} - {\x}^{\cc^-} \in \k[x_1, ..., x_n]$ as $\cc = \cc^+ - \cc^- \in \Z^n$ varies over the circuits of $\A$. This ideal is contained in the toric ideal, $\ia$, of $\A$ which has numerous applications and is nontrivial to compute. Since circuits can be computed using linear algebra and the two ideals often coincide, it is worthwhile to understand when equality occurs. In this paper we study $\ica$ in relation to $\ia$ from various algebraic and combinatorial perspectives. We prove that the obstruction to equality of the ideals is the existence of certain polytopes. This result is based on a complete characterization of the standard pairs/associated primes of a monomial initial ideal of $\ica$ and their differences from those for the corresponding toric initial ideal. Eisenbud and Sturmfels proved that $\ia$ is the unique minimal prime of $\ica$ and that the embedded primes of $\ica$ are indexed by certain faces of the cone spanned by $\A$. We provide a necessary condition for a particular face to index an embedded prime and a partial converse. Finally, we compare various polyhedral fans associated to $\ia$ and $\ica$. The Gröbner fan of $\ica$ is shown to refine that of $\ia$ when the codimension of the ideals is at most two.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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