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Huaiqing Zuo

Publications and source records attributed to Huaiqing Zuo.

At least 19 recordsLinked to original sources
arXiv2026

Non-existence of negative weight derivations on Jacobian algebras

Let $S=\mathbb K[x_1,\cdots,x_n]$ be a polynomial ring in $n$ variables with positive integer weights over a field $\mathbb K$ of characteristic zero. Let $f\in S$ be a weighted homogeneous polynomial such that the Jacobian algebra $$A=\left.S \middle/ \left(\frac{\partial f}{\partial x_1},\cdots,\frac{\partial f}{\partial x_n}\right)\right.$$ is a finite-dimensional $\mathbb K$-vector space. We prove that $A$ has no nonzero negative weight derivation, which confirms a conjecture of the third author. An immediate consequence of this conjecture is the micro-local characterization of quasi-homogeneous singularities.

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arXiv2026

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

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 and the Tjurina numbers for isolated hypersurface singularities of any dimension in positive characteristic. Using this result, we also derive an upper bound for the quotient of the Milnor numbers $μ$ and the Tjurina numbers $τ$ for isolated hypersurface singularities of any dimension in characteristic zero. In particular, as a corollary, we obtain that for an isolated surface singularity $(f,0) \subset (\mathbb{C}^3,0)$, $\frac{μ(f)}{τ(f)}< \frac{3}{2}$, which partially answers a conjecture of P. Almirón. This is also a weak version of Durfee's conjecture. We have also constructed a family of hypersurface singularities of any dimension for which $\fracμτ$ tends to the bound we get, which means that the bound is sharp, and at the same time answers an open problem raised by P. Almirón.

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arXiv2026

Nakai conjectures for isolated weighted homogeneous hypersurface singularities

The long-standing Nakai Conjecture asks whether differential operators detect singularities on algebraic varieties. On a smooth variety over a field of characteristic zero, the ring of differential operators is generated by first order derivations. Nakai asked whether the converse holds. In this paper, we prove that for the coordinate ring of an isolated weighted homogeneous hypersurface singularity, there always exists a second order derivation that is not generated by first order derivations. This verifies the Nakai Conjecture for isolated weighted homogeneous hypersurface singularities.

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arXiv2026

Analytic Criteria for Detecting Bernstein-Sato Roots

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 polynomial of an ideal. For homogeneous case, we recast this criterion as an analytic condition, connecting the algebraic problem of locating roots with analytic properties of the defining data. As an application, we give a series of roots for the general determinantal ideal prove the strong monodromy conjecture for them. Keywords. Bernstein-Sato polynomial, determinantal variety, monodromy conjecture.

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arXiv2026

The Hassanzadeh-Nasrollah Nejad-Simis Conjecture on Euler Conductors

We prove the Hassanzadeh-Nasrollah Nejad-Simis conjecture: if $f\in (x_1,\ldots,x_n)^2\subsetneq k[[x_1,\ldots,x_n]]$ has an isolated critical point and $\operatorname{char}k=0$, then its Euler conductor $J_f:f$ is not contained in the Tjurina ideal $(J_f,f)$. More generally, if $A=R/I$ is a nonzero Noetherian $k$-algebra and $a=[f]\in A$ is nilpotent with $d_{A/k}a=0$, then $I:f\nsubseteq(I,f)$. For a local ring $(R,\mathfrak n)$ we obtain the stronger noncontainment $I:f\nsubseteq(I,f)+\mathfrak n(I:f)$. The proof reduces a hypothetical containment to a self-exact square-zero element and detects its differential by the trace of a regular representation over the dual numbers. Finally, an explicit five-variable isolated singularity satisfies $J_f:f\subseteq\overline{J_f}$, disproving the integral-closure strengthening proposed by Ma and Zuo.

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arXiv2026

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

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 investigate their behavior under Thom--Sebastiani sums. For separated sums, we obtain a filtered defect formula and, in particular, give an affirmative answer to the question raised by Almirón concerning the Thom--Sebastiani property of the Tjurina subspectrum: it holds if and only if the Tjurina number is multiplicative. We also discuss rational sampling, spectral reconstruction, symmetry bounds, and examples illustrating the information captured and lost by integer Hodge--Tjurina data.

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arXiv2026

On the Tensor Property of Bernstein-Sato Polynomial

We prove the multiplicative Thom-Sebastiani rule for Bernstein-Sato polynomials, answering the longstanding questions of Budur and Popa. We generalize the result to the tensor of two effective divisors on the product of two arbitrary non-singular complex varieties. This also leads to a multiplicative property related to Igusa's strong monodromy conjecture. Moreover, we propose an extension of our result to Bernstein-Sato polynomials for ideals and prove it for monomial ideals.

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arXiv2026

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

Let $I\subseteq\mathbb C[x_1,\ldots,x_n]$ be an ideal generated by homogeneous polynomials of a common degree $d$. We give a polynomial partial differential equation criterion guaranteeing that $-n/d$ is a root of the Bernstein-Sato polynomial $b_I(s)$. We apply this criterion to the ideal of maximal minors of a generic $m\times n$ matrix and obtain the distinguished root $-n$; combined with local divisibility along determinantal strata, this allows us to obtain the strong monodromy conjecture in the maximal-minor case. Finally, we prove that the criterion is stable under enlarging the linear span of the generators, adjoining generators in disjoint variables, products satisfying the natural slope condition, and Thom-Sebastiani sums. These stability results provide new classes of homogeneous ideals and polynomials for which the distinguished Bernstein-Sato root can be detected. Keywords. Bernstein-Sato polynomial, monodromy conjecture.

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arXiv2026

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

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 subspace of the filter space. For one-dimensional finite-stride convolution, the filter-to-convolution-operator correspondence gives an injective linear map $\mathcal{C}:\mathcal{K}\to H$. This map sends filter subspaces in $\mathrm{Gr}(q,\mathcal{K})$ to operator subspaces in $\mathrm{Gr}(q,H)$; composing it with the Plücker embedding yields a projective parametrization $Φ:\mathrm{Gr}(q,\mathcal{K})\to\mathbb{P}(\bigwedge^q H)$. Using $T_U\mathrm{Gr}(q,\mathcal{K})\cong\mathrm{Hom}(U,\mathcal{K}/U)$, we compute the differential of the induced Grassmannian map and show that the differential of $Φ$ is injective at every point. We then use the vanishing equations for Plücker coordinates and standard affine coordinates on a Grassmannian to prove that $\mathrm{Gr}(q,\mathcal{C}(\mathcal{K}))\hookrightarrow\mathrm{Gr}(q,H)$ is a closed embedding, and hence that $Φ$ is a closed embedding. Consequently, the parameter space is isomorphic to its projective image, the parametrization is finite and birational onto its image, every fiber is a singleton, and the resulting projective neural variety is smooth. For $k=4$ and $q=2$, we also use Singular to recover the image ideal and check its dimension, degree, chart rank, and smoothness. This computation illustrates, rather than replaces, the general proof. Finally, we discuss possible connections with filter redundancy and low-rank convolution, while distinguishing the proved geometric results from application proposals requiring numerical validation.

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arXiv2026

Classification of unimodal isolated complete intersection singularities in positive characteristic

In this paper we classify the unimodal isolated complete intersection singularities in arbitrary characteristic under contact equivalence. The classification over $\mathbb{C}$ has already done by A. Dimca and C.G. Gibson. We continue and generalize their work. To complete the classification, we generalized the complete transversal method into positive characteristic field, which is also useful in many other classification problem.

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arXiv2026

Tjurina Number Jumps and Unimodal Hypersurface Singularities in Positive Characteristic

This paper generalizes existing methods to derive stronger bounds on the modality of hypersurface singularities. Our results demonstrate that each sudden jump in the extended Tjurina number necessarily increases the modality. Furthermore, we provide a full classification of unimodal isolated hypersurface singularities in characteristic p > 3 under contact equivalence.

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arXiv2026

Motivic principal value integrals for hyperplane arrangements

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 stronger positive answer to this conjecture for hyperplane arrangements.

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arXiv2026

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

Spectrum is an important numerical invariant of an isolated hypersurface singularity, connecting its topological and analytic structures. The well-known Hertling conjecture tells the relation of range and variance of exponents i.e. elements of spectrum. For trimodal singularities, we compute their spectra and verify Hertling conjecture for them. Jung, Kim, Saito and Yoon recently defined Tjurina spectrum, stemming from Hodge ideals. This set of numerical invariants is a subset of spectrum in Steenbrink's sense. We give an estimation of exponents not in Tjurina spectrum and propose a similar Generalized Hertling Conjecture for Tjurina Spectrum. Moreover, we prove the conjecture for singularities of modality $\leq 3$.

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