边界基概型的最优再嵌入
Optimal Re-Embeddings of Border Basis Schemes
AI总结：
本文研究边界基概型的最优再嵌入问题，通过构造检验候选元组的新算法并利用线性部分的Gröbner扇，改进了$Z$-分离再嵌入的计算技术，并应用于特定边界基概型。
摘要：
边界基概型是Hilbert概型的开子概型，用于参数化给定长度的$\mathbb{P}^n$的0维子概型。它们产生开覆盖，且易于描述和计算。我们的主题是寻找边界基概型到极小维仿射空间的再嵌入。给定$P = K[X] = K[x_1,\dots,x_n]$，理想$I\subseteq \langle X \rangle$，以及一组未定元$Z$，在先前的论文中，作者开发了计算$I$的$Z$-分离再嵌入的技术，即同构$Φ: P/I \rightarrow K[X\setminus Z] / (I\cap K[X\setminus Z])$。本文通过构造用于检验候选元组$Z$的新算法，并有效利用$I$的线性部分的Gröbner扇，进一步发展和改进了这些通用技术。然后我们将其应用于定义边界基概型$\mathbb{B}_{\mathcal{O}}$的理想，其中$\mathcal{O}$是项的序理想，以及它们的自然生成多项式。这些理想关于箭头分次是齐次的，这一事实使我们能够更系统地寻找合适的元组$Z$。利用未定元模极大理想的平方的等价性，我们快速计算出理想的线性部分的Gröbner扇，并在寻找最优再嵌入时确定哪些未定元应在$Z$中。具体应用包括$\mathcal{O}\subseteq K[x,y]$以及$\mathcal{O}$由直到某一阶数的所有项组成的边界基概型的再嵌入。
英文摘要：
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.
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。
