Huaiqing Zuo

Analytic Criteria for Detecting Bernstein-Sato Roots

Yifan Chen, Huaiqing Zuo

We study roots of Bernstein-Sato polynomials of ideals. Our first main result gives a general partial differential equation criterion for detecting roots of the Bernstein-Sato poly…

A combinatorial criterion for detecting roots of Bernstein-Sato polynomials

Quan Shi, Huaiqing Zuo

We give a combinatorial criterion for detecting roots of Bernstein-Sato polynomials of ideals. As applications, we prove the strong monodromy conjecture for determinantal and Pfaff…

math.AC2026

The Hassanzadeh-Nasrollah Nejad-Simis Conjecture on Euler Conductors

Yizhi Zhang, Huaiqing Zuo

We prove the Hassanzadeh-Nasrollah Nejad-Simis conjecture: if $f \in (x_{1}, \dots, x_{n})^{2} ⊊ k [[x_{1}, \dots, x_{n}]]$ has an isolated critical point and $char k = 0$ , t…

Hodge--Tjurina Numbers, Tjurina Spectra, and Thom--Sebastiani Formulas

Shijie Zheng, Huaiqing Zuo

We study Hodge--Tjurina numbers and the Tjurina subspectrum of isolated hypersurface singularities. We relate these invariants to Hodge ideals and microlocal $V$ -filtrations and in…

Zero loci of Bernstein-Sato ideals are not necessarily linear

Quan Shi, Yongxin Xu, Huaiqing Zuo

We find a polynomial map the zero locus of whose Bernstein-Sato ideal is not linear, giving a counterexample to a conjecture of N. Budur.

A Polynomial PDE Criterion for the $- n / d$ Root of the Bernstein-Sato polynomial of Homogeneous Ideals

Yifan Chen, Huaiqing Zuo

Let $I \subseteq C [x_{1}, \dots, x_{n}]$ be an ideal generated by homogeneous polynomials of a common degree $d$ . We give a polynomial partial differential equation criterion guar…

Grassmann--Plücker Parametrization of Convolutional Filter Subspaces: Regularity and Closed Embeddings

Hongyu Yuan, Huaiqing Zuo

We propose a geometric parametrization of the filters in a single convolutional layer: the parameter is no longer an ordered family of filter vectors, but a fixed-dimensional subsp…

On the Multiplicative Thom-Sebastiani Property for Bernstein-Sato Polynomials of Ideals

Yongxin Xu, Huaiqing Zuo

We prove that the Bernstein-Sato polynomials of ideals satisfy the multiplicative Thom-Sebastiani property. This affirmatively settles a question of Shi and Zuo that extends a ques…

The Quotient of Milnor Number by Tjurina Number of Hypersurface Singularities in Arbitrary Characteristic

Hongrui Ma, Huaiqing Zuo

In this paper, we use Hilbert-Samuel multiplicity, Hilbert-Kunz multiplicity, and s-multiplicity to establish a sharp upper bound for the quotient of the generalized Milnor numbers…

Spectrum, Tjurina spectrum, and Hertling conjecture for singularities of modality $\leq 3$

Quan Shi, Yang Wang, Huaiqing Zuo

Spectrum is an important numerical invariant of an isolated hypersurface singularity, connecting its topological and analytic structures. The well-known Hertling conjecture tells t…

A remarkable subset of poles of the motivic zeta function

Nero Budur, Eduardo de Lorenzo Poza, Quan Shi +1

For any polynomial f with complex coefficients we find a remarkable subset of poles of the motivic zeta function. It is combinatorially determined by any log resolution and it admi…

On the Poles of Real Archimedean Zeta Functions

Zhikuang Chen, Huaiqing Zuo

This paper studies the poles of the real Archimedean zeta function for a weighted homogeneous polynomial $f \in R [x, y]$ with an isolated singularity at the origin. By app…

On the monodromy conjecture, holomorphy conjecture, and embedded Nash problem for Pfaffian ideals

Yifan Chen, Quan Shi, Yongxin Xu +1

We resolve the monodromy conjecture, holomorphy conjecture, and embedded Nash problem for Pfaffian ideals.

On the monodromy conjecture for determinantal varieties

Yifan Chen, Huaiqing Zuo

This paper presents a proof of the monodromy conjecture for determinantal varieties. Our strategy centers on an in-depth analysis of monodromy zeta functions, leveraging a generali…

math.AC2025

On jet closures of singularities

Yifan Chen, Huaiqing Zuo

The jet closure and jet support closure were first introduced by de Fernex, Ein and Ishii to solve the local isomorphism problem. In this paper, we introduce two local algebras ass…

Hilbert series of second order jets of determinantal varieties

Yifan Chen, Yongxin Xu, Huaiqing Zuo

In this paper, we will investigate the jet schemes of determinantal varieties. It is quite often the case that the geometric information concerning the jet schemes of an algebraic…

On jet schemes of determinantal varieties

Yifan Chen, Huaiqing Zuo

Determinantal varieties are important objects of study in algebraic geometry. In this paper, we will investigate them using the jet scheme approach. We have found a new connection…

Polar loci of multivariable archimedean zeta functions

Nero Budur, Quan Shi, Huaiqing Zuo

We determine, up to exponentiating, the polar locus of the multivariable archimedean zeta function associated to a finite collection of polynomials F. The result is the monodromy s…

On Stringy E-functions and the Non-negativity Conjecture for Determinantal Varieties

Yifan Chen, Huaiqing Zuo

We compute the stringy E-functions of determinantal varieties and establish that the stringy E-function of a determinantal variety coincides with the E-function of the product of a…

On Motivic Zeta Functions and Stringy E-function via Embedded $Q$ -Resolution

Yifan Chen, Quan Shi, Huaiqing Zuo

We provide the formula of motivic zeta function for semi-quasihomogeneous singularities and in dimension two, we determine the poles of zeta functions. We also give another formula…

Motivic principal value integrals for hyperplane arrangements

Nero Budur, Quan Shi, Huaiqing Zuo

A conjecture of Denef-Jacobs-Veys relates motivic principal value integrals of multivalued rational top-forms with cohomology support loci of rank one local systems. We give a stro…

Variation of Archimedean Zeta Function and $n / d$ -Conjecture for Generic Multiplicities

Quan Shi, Huaiqing Zuo

For $f_{1}, ..., f_{r} \in C [z_{1}, ..., z_{n}] ∖ C$ , we introduce the variation of archimedean zeta function. As an application, we show that the $n / d$ -conjecture, propo…

math.AC2024

Bernstein-Sato functional equations for ideals in positive characteristic

Siyong Tao, Zida Xiao, Huaiqing Zuo

For an ideal of a regular $\cc$-algebra, its Bernstein-Sato polynomial is the monic polynomial of the lowest degree satisfying an Bernstein-Sato functional equation. We generalize…