設 (R, m) 為一個完備局部noetherian Z/(p) -代數，m[n] 是ｍ的第Pn個Frobenius乘冪，即m[n] 是由ｍ的元素之Pn次方所生成的理想。令en(R) 代表R/m[n] 的長度，函數n|→en (R) 則稱為Ｒ的 Hilbert-Kunz函數。
設Ｆ是一個特徵p > 0的體，S＝F〔|S1,...,Xr,Y1,...,Ys,Z1,...,Zt|〕。在本論文中，利用Groebner基底，我們探討二項超曲面之Hilbert-Kunz函數：
f：＝X1(a1...Xr(ar))Y1(b1)...Ys(bs)＋Y1(c1)...Ys(cs)Z1(d1))...Zt(dr)
其中，對於每一個k，bk≧ck。令R＝S/<f>，在第二節我們討論r≧1的情形；而在第三節則討論r＝0的情形。在命題2.7和3.7，我們精確地找出Ｒ的Hilbert-Kunz函數，而在定理2.8和3.8，當ｎ充分大時，我們證明了這些fk (n)均是ｎ的週期函數。
至於一般的二項超曲面，要精確找出Hilbert-Kunz函數，那就更複雜了。

Let (R,m) be a complete local noetherian Z/(p)-algebra, and m[n] the Pn-th Frobenius power of m, i.e. the ideal generated by the Pn -th powers of the elements of m . Let en(R) be the length of R/m[n]. The function n|→en (R) is called the Hilbert-Kunz function of R.
Let F be a field of characteristic p ＞ 0 and S＝F〔|S1,...,Xr,Y1,...,Ys,Z1,...,Zt|〕. In this article, by making use of Froebner basis, we determine the Hilbert-Kunz function of binomial hypersurfaces of the form:
f：＝X1(a1...Xr(ar))Y1(b1)...Ys(bs)＋Y1(c1)...Ys(cs)Z1(d1))...Zt(dr)
with bk≧ck for each k. LetR＝S/<f>，. In Section 2, we discuss the case r≧1 and the discussion for r＝0 is in section 3. In Proposition 2.7 and 3.7, we explicitly determine the Hilbert-Kunz function, and in Theorem 2.8 and 3.8, we show that for n>>0, where λ is a rational number and fk(n) is an eventually periodic function of n for each k.
For the general binomial hypersufaces, it is more complicated to explicitly determine the Hibert-Kunz function.