Furthermore, if l = 1, then λ is an integer and f (n) is an eventually periodic function of n for n ≫ 0. In other situation, it is not known about λ and f (n).

1 Introduction

Let p be a prime, (R, M) be a complete local Noetherian Z/(p)-algebra and M ^[n] be the p ⁿ -th Frobenius power of M, that is the ideal generated by the p ⁿ -th power of the elements of M. Let HK R (n) be the length of R/M ^[n] . The function n 7−→ HK R (n) is “the Hilbert-Kunz function” of R.

Suppose that the Krull dimension, l, of R is > 0. In [10] Monsky showed that there exists a positive real constant λ and a function f (n) = O(p ^(l−1)n ) such that

HK R (n) = λp ^ln + f (n).

Furthermore, if l = 1, then λ is an integer and f (n) is an eventually periodic function of n for n ≫ 0. In other situation, it is not known about λ and f (n).

In general, it is hard to determine the Hilbert-Kunz function of a ring. However, there are already some results for some classes of special rings, for example, [1, 2, 3, 4, 6, 7, 8, 9, 10, 11].

In [8] Kunz determined the Hilbert-Kunz functions when

R = Z/(p) [X 1 , . . . , X s ] ,

X ₁ ^d

−

s

Y

i=2

X _i ^d

! ,

where s = l + 1 ≥ 3, and showed that λ is rational and f (n) =

l−1

X

k=0

f k (n)p ^kn with each f k

eventually periodic.

In [6] Han and Monsky determined the Hilbert-Kunz function when

R = Z/(p) [X 1 , . . . , X s ]

, _s X

i=1

X _i ^d

! ,

where the d i are positive integers and s = l + 1 ≥ 3. They proved that λ is rational and f (n) is an eventually periodic function of n for n ≫ 0 when p = 2 or s = 3. They also showed that if p > 2 and s > 3, there is a rational λ and integers l and µ, µ ≥ 1, 0 ≤ l ≤ p ^(s−3)µ such that f (n + µ) = l · f (n) for n ≫ 0.

In [4] Conca determined the Hilbert-Kunz functions when R = K[X 1 , . . . , X s ]/I

where K is a field of characteristic p > 0 and I is a monomial ideal or a principal ideal generated by a homogeneous binomial form X ^a − X ^b where the greatest common divisor of X ^a and X ^b is 1.

In [3] Chiang and Hung determined the Hilbert-Kunz functions of all hypersurfaces of the form

f :=

m

X

i=1 t

Y

j=1

X _ij ^d

with d ij ≥ 1.

1

In [9] Lin determined the Hilbert-Kunz functions when

R = K[X 1 , . . . , X r , Y 1 , . . . , Y s , Z 1 , . . . , Z t ]/I

where K is a field of characteristic p > 0 and I is a general binomial ideal.

In this paper, by making use of Gr¨obner basis, we determine some cases of the Hilbert- Kunz function of trinomial hypersurfaces of the form

f := X ^a Y ^b + Y ^c Z ^d + Z ^e with 0 < a ≤ b ≤ c.

In Section 2, we arrange the notations and review some definitions.

In Section 3, we discuss the case that d ≥ e, and we obtain the following theorem.

Theorem 1.1 The Hilbert-Kunz function of the hypersurface X ^a Y ^b + Y ^c Z ^d + Z ^e with 0 < a ≤ b ≤ c and d ≥ e is

n 7−→ λp ²ⁿ + f 1 (n)p ⁿ + f 0 (n)

for n ≫ 0, where λ =

" ₂ X

k=1

(−1) ^k+1 S k (a, b) e u ^k

#

, u = max{b, e}, and f k (n) is an eventually periodic function of n for each k.

In Section 4, we discuss the other cases and obtain the same conclusions as that in Theorem 1.1 with the same method.

2