AI 中文总结
本文研究自由超平面排列的Bernstein-Sato理想，借助AI计算了两族理想，推广了已有公式，并给出反例反驳了Budur的猜想。
AI 中文摘要
设$f=(f_1,\dots,f_r)$为$X=\mathbb{C}^n$中中心超平面排列$D$的完全分解。对于幺半群理想$K\subseteq \mathbb{N}^r$，我们研究$f$沿$K$的Bernstein-Sato理想$B^K_f$，即$\mathcal{D}_X[s]f^s/\sum_{m\in K}\mathcal{D}_X[s]f^{s+m}$的$\mathbb{C}[s]$-零化子。当$D$自由时，我们借助AI计算了这些理想的两个族。对于单位平移$K=\langle e_i\rangle$，我们证明$B^{-e_i}_f$由$D$中包含于$D_i$的稠密边索引的线性形式的显式乘积生成。这确定了自由排列的所有Bernstein-Sato理想$B^{a,b}_f=\operatorname{Ann}_{\mathbb{C}[s]}\mathcal{D}_X[s]f^{s-a}/\mathcal{D}_X[s]f^{s-b}$，$a\geq b$，推广了Maisonobe（2016）和Bath（2020）的公式。主要的新工具将$\mathcal{D}_X[s]f^s/\mathcal{D}_X[s]f^{s+e_i}$沿原点余法丛的相对特征簇的重数识别为与$D$的通用Ziegler限制相关的Artinian完全交的Hilbert级数的系数；Wu（2022）计算的总重数迫使所有得到的系数上界成为等式。对于坐标幺半群理想$K=\langle e_1,\dots,e_r\rangle$，我们证明$B^K_f$由$D$的本质商的每个不可约因子生成一个Euler关系。最后，我们证明沿幺半群理想的Bernstein-Sato理想的零点集不必是平移线性子簇的有限并，即使对于$\mathbb{C}^2$中的约化自由排列也是如此：对于$f=(x,y,x+y,x+2y)$和$K=\langle 3e_1,3e_2\rangle$，我们精确计算了$B^K_f$并发现了一个不可约二次分量。这反驳了Budur的一个猜想。
英文摘要
Let $f=(f_1,\dots,f_r)$ be a complete factorization of a central hyperplane arrangement $D$ in $X=\mathbb{C}^n$. For a monoid ideal $K\subseteq \mathbb{N}^r$ we study the Bernstein-Sato ideal $B^K_f$ of $f$ along $K$, that is, the $\mathbb{C}[s]$-annihilator of $\mathcal{D}_X[s]f^s/\sum_{m\in K}\mathcal{D}_X[s]f^{s+m}$. When $D$ is free we compute two families of these ideals with the help of AI. For the unit shift $K=\langle e_i\rangle$ we prove that $B^{-e_i}_f$ is generated by an explicit product of linear forms indexed by the dense edges of $D$ contained in $D_i$. This determines all the Bernstein-Sato ideals $B^{a,b}_f=\operatorname{Ann}_{\mathbb{C}[s]}\mathcal{D}_X[s]f^{s-a}/\mathcal{D}_X[s]f^{s-b}$, $a\geq b$, of a free arrangement, generalizing formulas of Maisonobe (2016) and Bath (2020). The main new ingredient identifies the multiplicities of the relative characteristic cycle of $\mathcal{D}_X[s]f^s/\mathcal{D}_X[s]f^{s+e_i}$ along the conormal bundle of the origin with the coefficients of the Hilbert series of an Artinian complete intersection attached to a generic Ziegler restriction of $D$; the total multiplicity computed in Wu (2022) then forces all the resulting coefficientwise upper bounds to be equalities. For the coordinate monoid ideal $K=\langle e_1,\dots,e_r\rangle$ we show that $B^K_f$ is generated by one Euler relation for each irreducible factor of the essential quotient of $D$. Finally, we show that the zero locus of a Bernstein-Sato ideal along a monoid ideal need not be a finite union of translated linear subvarieties, even for a reduced free arrangement in $\mathbb{C}^2$: for $f=(x,y,x+y,x+2y)$ and $K=\langle 3e_1,3e_2\rangle$ we compute $B^K_f$ exactly and find an irreducible quadric component. This disproves a conjecture due to Budur.
