论文标题
边境基础方案的最佳重新安装
Optimal Re-Embeddings of Border Basis Schemes
论文作者
论文摘要
边界基础方案是希尔伯特方案的开放式计算，参数化给定长度的$ \ mathbb {p}^n $的0维亚化。它们产生开放式覆盖物，易于描述和计算。我们的主题是将边界基方案的重新安置到最小维度的仿射空间中。给定$ p = k [x] = k [x_1，\ dots，x_n] $，一种理想的$ i \ subseteq \ langle x \ rangle $，以及一个不确定的元组$ z $，在以前的论文中，作者在作者中开发了用于计算$ z $ z $ z $ z $的$ i $ i $ i $ i $ i，i.eme的技术， \ rightarrow k [x \ setminus z] /（i \ cap k [x \ setminus z]）$。在这里，通过构建一种用于检查候选元组$ z $的新算法并通过使用$ i $的线性部分的gröbner粉丝，通过构建一种新算法来进一步开发这些通用技术。 然后，我们将其应用于定义边界基础方案$ \ Mathbb {b} _ {\ Mathcal {o}} $的理想，其中$ \ Mathcal {O} $是术语的订单理想，也是其自然产生的多项式。这些理想是均匀的W.R.T.的事实箭头分级使我们可以更系统地寻找合适的元组$ z $。使用不确定的Modulo的等效性最大理想的平方，我们快速计算了理想的线性部分的gröbner粉丝，并确定当我们寻找最佳重新安装时，哪些不确定的不确定性应为$ z $。特定的应用程序包括对边界基方案的重新安排，其中$ \ Mathcal {O} \ subseteq K [x，y] $以及$ \ Mathcal {o} $在其中包含所有条款。
Border basis schemes are open subschemes of Hilbert schemes parametrizing 0-dimensional subschemes of $\mathbb{P}^n$ of given length. They yield open coverings and are easy to describe and to compute with. Our topic is to find re-embeddings of border basis schemes into affine spaces of minimal dimension. Given $P = K[X] = K[x_1,\dots,x_n]$, an ideal $I\subseteq \langle X \rangle$, and a tuple $Z$ of indeterminates, in previous papers the authors developed techniques for computing $Z$-separating re-embeddings of $I$, i.e., of isomorphisms $Φ: P/I \rightarrow K[X\setminus Z] / (I\cap K[X\setminus Z])$. Here these general techniques are developed further and improved by constructing a new algorithm for checking candidate tuples $Z$ and by using the Gröbner fan of the linear part of $I$ advantageously. Then we apply this to the ideals defining border basis schemes $\mathbb{B}_{\mathcal{O}}$, where $\mathcal{O}$ is an order ideal of terms, and to their natural generating polynomials. The fact that these ideals are homogeneous w.r.t. the arrow grading allows us to look for suitable tuples $Z$ more systematically. Using the equivalence of indeterminates modulo the square of the maximal ideal, we compute the Gröbner fan of the linear part of the ideal quickly and determine which indeterminates should be in $Z$ when we are looking for optimal re-embeddings. Specific applications include re-embeddings of border basis schemes where $\mathcal{O}\subseteq K[x,y]$ and where $\mathcal{O}$ consists of all terms up to some degree.
