关于重数界的注记

某些Hilbert-Kunz重数的代数性（模猜想）

Algebraicity of some Hilbert-Kunz multiplicities (modulo a conjecture)

设F是特征为2的有限域，h是F[[x，y，z]]的元素x^3+y^3+xyz.在较早的文件中，我们提出了一个精确的猜想，以价值的colonies的理想（x^q，y^q，z^q，h^j）为q的幂2。我们还证明，如果猜想成立，那么H=uv+h的Hilbert-Kunz级数是Q（w）上的代数（2次），并且mu（h）是代数的（明确地，（4/3）+（5/14）sqrt（7））。在本文中，假设相同的猜想，我们使用无限矩阵理论重新推导这个结果，并将其推广到更广泛的H类，例如H=g（u，v）+h。在后续的论文中，在同样的假设下，我们将证明超越希尔伯特-昆兹多重性的存在。

Multiplicities and Mixed Multiplicities of arbitrary Filtrations

We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of $m$-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of $r$ filtrations on an analytically unramified local ring $R$ come from the coefficients of a suitable homogeneous polynomial in $r$ variables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities of $m$-primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterizat…

定维局部环中Hilbert-Kunz重数的下界

Lower bounds for Hilbert-Kunz multiplicities in local rings of fixed dimension

设$(R,\m)$是具有正素特征和维数$d$的形式非混合局部环.我们研究了具有小希尔伯特-昆兹多重性的含义（即，接近1）。特别是，我们表明，如果$R$是不规则的，存在一个下界，严格大于1，仅取决于$d$，其希尔伯特-昆兹多重性。

定维局部环的Hilbert-Kunz重数的新估计

New estimates of Hilbert-Kunz multiplicities for local rings of fixed dimension

给出了正特征局部环的Hilbert-Kunz重数的Watanabe-Yoshida猜想的结果.通过改进给出Hilbert-Kunz重数下界的“体积估计”，得到了当环的Hilbert-Samuel重数小于或等于5，或维数小于或等于6时的猜想.对于固定维数的非正则环，得到了Hilbert-Kunz重数的一个新的下界.

On the projective dimension of $5$ quadric almost complete intersections with low multiplicities

Let $S$ be a polynomial ring over an algebraic closed field $k$ and $ \mathfrak p =(x,y,z,w) $ a homogeneous height four prime ideal. We give a finite characterization of the degree two component of ideals primary to $\mathfrak p$, with multiplicity $e \leq 3$. We use this result to give a tight bound on the projective dimension of almost complete intersections generated by five quadrics with $e \leq 3$.

A transformation rule for natural multiplicities

For multiplicities arising from a family of ideals we provide a general approach to transformation rules for a ring extension étale in codimension one. Our result can be applied to bound the size of the local étale fundamental group of a singularity in terms of F-signature, recovering a recent result of Carvajal-Rojas, Schwede, and Tucker, and differential signature, providing the first characteristic-free effective bound.

The lower bound on the HK multiplicities of quadric hypersurfaces

Here we prove that the Hilbert-Kunz mulitiplicity of a quadric hypersurface of dimension $d$ and odd characteristic $p\geq 2d-4$ is bounded below by $1+m_d$, where $m_d$ is the $d^{th}$ coefficient in the expansion of $\mbox{sec}+\mbox{tan}$. This proves a part of the long standing conjecture of Watanabe-Yoshida. We also give an upper bound on the HK multiplicity of such a hypersurface. We approach the question using the HK density function and the classification of ACM bundles on the smooth quadrics via matrix factorizations.

Artin-Nagata性质、最小重数和纤维锥的深度

Artin-Nagata properties, minimal multiplicities, and depth of fiber cones

理想重数的极小值与爆破代数的深度有很强的关系。本文引入了极大解析展开理想的Goto-极小$j$ -重数的概念。在科恩-麦考利戒指，灵感来自S。Goto，A. Jayanthan，T. Puthenpurakal和J. Verma提出的最小$j$ -多重性的概念之间的相互作用。Polini和Y. Xie，以及满足某些剩余假设的纤维理想锥的Cohen-Macaulayness。我们还能够提供一个限制的减少Goto最小的$j$ -多重理想的数量，具有科恩-麦考利相关的分次代数，或线性衰减的权力的深度。

Lower bounds on Hilbert--Kunz multiplicities and maximal F-signatures

Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert--Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert--Kunz multiplicity.

理想的混合重数与多面体的混合体积

Mixed multiplicities of ideals versus mixed volumes of polytopes

本文的主要结果将格多面体的混合体积解释为理想的混合重数，将理想的混合重数解释为Samuel重数。特别是，我们可以给一个纯粹的代数证明伯恩斯坦定理断言，共同零点的数量系统的洛朗多项式方程在环面是有界的混合体积以上的牛顿多面体。

希尔伯特系数与萨利模：Vasconcelos贡献综述

Hilbert Coefficients and Sally Modules: A Survey of Vasconcelos' Contributions

本文是关于Wolmer Vasconcelos在重数、希尔伯特系数等相关不变量领域贡献的综述，重点介绍了其引入的萨利模这一连接希尔伯特系数与约化数的重要研究工具。

弱分次理想族的重数

Multiplicities of weakly graded families of ideals

本文推广线性下有界弱分次理想族的重数概念，证明相关极限存在性、体积重数公式与Minkowski不等式，探究特定形式弱分次理想族的重数性质、不等式取等条件，推广Rees的结果并研究相关长度函数的渐近行为。

边理想的广义重数

Generalized multiplicities of edge ideals

我们利用交换代数和多面体几何的方法，探讨了无平方单项式理想的广义重数与超图的组合结构之间的联系。例如，我们证明了连通超图的连通分支上的$j$ -重数是乘法的，并且明确地将适当连通的一致超图的边理想的$j$ -重数与其特殊纤维环的Hilbert-Samuel重数联系起来.此外，我们提供了一般的边界的广义重的边理想和计算这些不变量类的一致超图。

弱 Lefschetz 性质与单项式理想的混合重数

The weak Lefschetz property and mixed multiplicities of monomial ideals

本文利用等生成单项式理想的分析扩张、混合重数与双有理单项式映射的联系，以关联理想的混合重数给出平方自由约化满足弱 Lefschetz 性质的充要条件，并推广至正特征情形。

The Minkowski Equality for Filtrations

Suppose that R is an analytically irreducible or excellent local domain with maximal ideal m_R. We consider multiplicities and mixed multiplicities of R by filtrations of m_R-primary ideals. We show that the theorem of Teissier, Rees and Sharp, and Katz, characterizing equality in the Minkowski inequality for multiplicities of ideals, is true for divisorial filtrations, and for the larger category of bounded filtrations. This theorem is not true for arbitrary filtrations of m_R-primary ideals.

Bounds of the multiplicity of abelian quotient complete intersection singularities

In this paper, we investigate the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in term of the special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection singularities.

D-模上的滤子与Bernstein-Sato多项式根的重数

Filtrations on D-modules and multiplicities of roots of Bernstein-Sato polynomials

本文关联Bernstein-Sato多项式根的重数与奇点不变量，引入模b-函数并给出其算法，解决Gelfand问题的渐近解，还将结果应用于Torelli问题并给出肯定答案。