## Tokyo Journal of Mathematics

### A Function Determined by a Hypersurface of Positive Characteristic

Kosuke OHTA

#### Abstract

Let $R=k\jump{X_1, \dots ,X_{n+1}}$ be a formal power series ring over a perfect field $k$ of characteristic $p>0$, and let $\mathfrak{m} = (X_1 , \dots , X_{n+1})$ be the maximal ideal of $R$. Suppose $0\neq f \in\mathfrak{m}$. In this paper, we introduce a function $\xi_{f}(x)$ associated with a hypersurface $R/(f)$ defined on the closed interval $[0,1]$ in $\mathbb{R}$. The Hilbert-Kunz multiplicity and the F-signature of $R/(f)$ appear as the values of our function $\xi_{f}(x)$ on the interval's endpoints. The F-signature of the pair, denoted by $s(R,f^{t})$, was defined by Blickle, Schwede and Tucker. Our function $\xi_{f}(x)$ is integrable, and the integral $\int_{t}^{1}\xi_{f}(x)dx$ is just $s(R,f^{t})$ for any $t\in[0,1]$.

#### Article information

Source
Tokyo J. Math., Volume 40, Number 2 (2017), 495-515.

Dates
First available in Project Euclid: 9 January 2018

https://projecteuclid.org/euclid.tjm/1515466837

Mathematical Reviews number (MathSciNet)
MR3743730

Zentralblatt MATH identifier
06855946

Subjects

#### Citation

OHTA, Kosuke. A Function Determined by a Hypersurface of Positive Characteristic. Tokyo J. Math. 40 (2017), no. 2, 495--515. https://projecteuclid.org/euclid.tjm/1515466837

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