$F$-thresholds and test ideals of Thom-Sebastiani type polynomials

关于理想的Bernstein-Sato多项式的乘性Thom-Sebastiani性质

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

该研究证明理想的Bernstein-Sato多项式满足乘性Thom-Sebastiani性质，正面解决了相关领域的一个公开问题

The Tjurina number for Sebastiani-Thom type isolated hypersurface singularities

In this note we provide a formula for the Tjurina number of a join of isolated hypersurface singularities in separated variables. From this we are able to provide a characterization of isolated hypersurface singularities whose difference between the Milnor and Tjurina numbers is less or equal than two arising as the join of isolated hypersurface singularities in separated variables. Also, we are able to provide new upper bounds for the quotient of Milnor and Tjurina numbers of certain join of isolated hypersurface singularities. Finally, we deduce an upper bound for the quotient of Milnor and …

正特征下理想的 Bernstein–Sato 函数方程

Bernstein-Sato functional equations for ideals in positive characteristic

本文将 Bernstein–Sato 函数方程推广至正特征 F-有限环中的理想，刻画其与 Bernstein–Sato 根的关系，并给出加权齐次孤立奇点多项式的根及 Thom–Sebastiani 性质。

齐次理想的Bernstein-Sato多项式的$-n/d$根的多项式PDE准则

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

针对由次数为d的齐次多项式生成的理想，给出判定其Bernstein-Sato多项式$-n/d$根的多项式PDE准则，应用于极大子式理想得特殊根$-n$，且该准则具稳定性。

The integral closure of a primary ideal is not always primary

In 1936, Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic $p=2$. We provide counterexamples to Krull's question regarding polynomial rings with any characteristics. We also find that the Jacobian ideal $J$ of the polynomial $f = x^6 + y^6 + x^4 z t + z^3$ given by Briançon and Speder in 1975 is a counterexample to Krull's question. Let $V_1$ be the hypersurface defined by $f = 0$ and $V_2$ be its …

多项式理想的合冲基

Involutive Bases of Polynomial Ideals

本文提出一种基于合冲单项式除法的通用算法，用于构造多项式理想的合冲基，并证明其为Gröbner基的特殊冗余形式，同时引入优化策略减少计算量。