Bounds for Codimension-One Components of Zero Loci of Bernstein-Sato Ideals

Abstract

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 $c\leq L_E(\mathbf a)+(n-1-\delta_f)L_E(\mathbf 1)-k_E$, where $\delta_f=\min\{n-1,\alpha_f\}$ and $\alpha_f$ 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.