极限 $F$-签名函数：二元二项式超曲面

F-pure thresholds of binomial hypersurfaces

In this note, we use estimates given in the recent preprint [Her11b]to deduce a formula for the F-pure threshold of a binomial hypersurface over a field of prime characteristic. These formulas are given in terms of the associated splitting polytope, and remain valid over any characteristic.

The Hilbert-Kunz function for Binomial Hypersurfaces

In this article, I give an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general, over any feild of arbitrary positive characteristic. I prove that the Hilbert-Kunz multiplicity associated to any Binomial Hypersurface over any field of arbitrary positive characteristic is rational. As an example, I also prove the well known fact that for 1-dimensional Binomial Hypersurfaces, the Hilbert-Kunz multiplicity is a positive integer and give a precise account of the integer.

FFRT Properties of hypersurfaces and their F-signature

This paper studies properties of certain hypersurfaces in prime characteristic: we give a sufficient and necessary conditions for some classes of such hypersurfaces to have Finite $F$-representation Type (FFRT) and we compute the $F$-signatures of these hypersurfaces. The main method used in this paper is based on finding explicit matrix factorizations.

对角超曲面的 F-signature 函数

F-signature functions of diagonal hypersurfaces

研究对角超曲面的F-signature函数，证明其随素数趋于无穷收敛于分段多项式函数，并用于证明极限F-signature和Hilbert-Kunz重数的存在性，对Fermat超曲面给出显式公式并否定回答Watanabe-Yoshida的问题。

简单奇点与费马超曲面的极限F-不变量

Limit F-invariants of the simple singularities and the Fermat hypersurfaces

本文计算了简单奇点与费马超曲面的极限F-不变量，推广了相关公式，并将正则环刻画与Watanabe--Yoshida猜想推广至特征零。

A function determined by a hypersurface of positive characteristic

Let R be a formal power series ring over a perfect field k of prime characteristic p, and let m be the maximal ideal of R. Suppose f is a non-zero element in m. In this paper, we introduce a function xi (x) associated with a hypersurface defined on the closed interval [0, 1] in R. The Hilbert-Kunz function and the F-signature of a hypersurface appear as the values of our function xi (x) on the interval s endpoints.