本文之主要目的為利用代表環(representation rings)這種工具來得
到某些超曲面 座標環的 Hilbert-Kunz 函數之一般式. 主要結果如
下:

特徵數為 $p$ 之體中, 超曲面

$f:=(X_1 Y_1)^{d_1}+(X_2 Y_2)^{d_2}+..+(X_s Y_s)^{d_s}+{X_{s+1
}}^{d_{s+1}}+.. +{X_t}^{d_t}$ 之 Hilbert-Kunz 函數是

\[

n \mapsto cp^{n(t+s-1)}+\Delta_1(n)+\Delta_2(n)+..+\Delta_{2^s}(
n), \]

其中 $c$ 是一個有理數, 而 $\Delta_i(n)$ 之性質如下:

存在正整數 $\rho$, $\omega$, $l_i$, 整數函數 $\tau$, $\kappa$:
(定義域,值 域都在整數)

\[

n=\rho+1+\tau(n)\omega+\kappa(n)$,

\]

這裡 $0 \le \kappa(n) < \omega$, 以及週期為 $\omega$ 之函數
$\phi_i$, 使得當 $n > \rho$ 時

\[

\Delta_i(n)={l_i}^{\tau(n)}+\phi_i(n),

\]

其中 $l_$ 之上界為 $p^{\omega(t+s-3)}$.

In this article, we use the tool "representation rings" to
obtained someHilbert-Kunz functions of the coordinate rings of
some hypersurfaces. The mainresults is as follows: If the
characteristic numberof the fieldis $p >0$, the Hilbert-
Kunzfunction of the coordinate ring of the hypersurface $f:=(X_1
Y_1)^{d_1}+(X_2 Y_2)^{d_2}+..+(X_s Y_s)^{d_s}+{X_{s+1}}^{d_{s+1
}}+..+{X_t}^{d_t}$ is \[ n \mapsto cp^{n(t+s-1)}+\Delta_1(n)+
\Delta_2(n)+..+\Delta_{2^s}(n),\]where $c$ is rational, and
$\Delta_i(n)$ has the following properties: There is positive
integers $\rho$, $\omega$, $l_i$, functions $\tau$, $\kappa$:
\[n=\rho+1+\tau(n)\omega+\kappa(n)$,\]where $0 \le \kappa(n) <
\omega$, and periodic function $\phi_i$ with period $\omega$,
such that$n > \rho$ \[\Delta_i(n)={l_i}^{\tau(n)}+\phi_i(n)\]as
$n > \rho$the upper bound of $l_i$ is $p^{\omega(t+s-3)}$.
In this article, we use the tool "representation rings" to