发表机构
School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究 Bernstein-Sato 理想零轨迹的余一维分量，证明其超平面截距的上界，解决了 Budur 等人提出的任意元组情形的问题，并给出新证明与除数值表述。
AI 中文摘要
设 $X$ 为 $n$ 维光滑复仿射簇，$F=(f_1,\ldots,f_r)$ 为 $X$ 上非零正则函数构成的元组，且 $f:=\prod_{i=1}^r f_i$ 不可逆。我们研究非负整数平移 $\mathbf a$ 下 Bernstein-Sato 理想 $B_F^{\mathbf a}$ 的零轨迹。对于固定的对数消解，$Z(B_F^{\mathbf a})$ 的每个余一维不可约分量均为形如 $L_E(\mathbf s)+k_E+c=0$ 的超平面，其中 $c$ 为正整数。我们利用相对 $D$-模的局部化极大与极小扩张给出了该结果的新证明。我们还证明了 $c\leq L_E(\mathbf a)+(n-1-\delta_f)L_E(\mathbf 1)-k_E$，其中 $\delta_f=\min\{n-1,\alpha_f\}$，$\alpha_f$ 为 $f$ 的极小指数。对任意元组（特别是 $r>1$ 情形）获得此类上界的问题由 Budur、van der Veer 和 Van Werde 提出，上述不等式解决了该问题。为获得上界，我们将 $Z(B_F^{\mathbf 1})$ 的对角切片与 $b_f$ 的根集进行比较。通过平移的有限覆盖，结合对角特化与对数消解描述，表明这些集合具有相同的最小点和最大点。Saito 在其共同最小点处的根估计进而给出上界。我们进一步建立了局部指标比较的除数值表述，恢复了通过单值 ζ 函数与多变量 A'Campo 公式检测单值支集的结果。
英文摘要
Let $X$ be a smooth complex affine variety of dimension $n$, and let $F=(f_1,\ldots,f_r)$ be a tuple of nonzero regular functions on $X$ such that $f:=\prod_{i=1}^r f_i$ is not invertible. We study the zero loci of the Bernstein-Sato ideals $B_F^{\mathbf a}$ for nonnegative integral shifts $\mathbf a$. For a fixed log resolution, every codimension-one irreducible component of $Z(B_F^{\mathbf a})$ is a hyperplane of the form $L_E(\mathbf s)+k_E+c=0$ with $c$ a positive integer. We give a new proof of this result using localized maximal and minimal extensions of relative D-modules. We also prove that $c\leq L_E(\mathbf a)+(n-1-δ_f)L_E(\mathbf 1)-k_E$, where $δ_f=\min\{n-1,α_f\}$ and $α_f$ is the minimal exponent of $f$. The problem of obtaining such an upper bound for arbitrary tuples (in particular, for $r>1$) was raised by Budur, van der Veer, and Van Werde, and the above inequality resolves it. To obtain the upper bound, we compare the diagonal slice of $Z(B_F^{\mathbf 1})$ with the root set of $b_f$. A finite covering by translates, combined with diagonal specialization and the log-resolution description, shows that these sets have the same least and greatest points. Saito's root estimate at their common least point then yields the upper bound. We further establish a divisor-valued formulation of the local index comparison, recovering the detection of monodromy support via monodromy zeta functions and the multivariable A'Campo formula.
