In this section we shall use the same method of Section 3 to determine the other situations of the Hilbert-Kunz function of the hypersurfaces of the following form:

4 The other cases

Let K be a field of characteristic p > 0 and

S = K[X, Y, Z].

In this section we shall use the same method of Section 3 to determine the other situations of the Hilbert-Kunz function of the hypersurfaces of the following form:

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

Let q = p ⁿ and set R = S/hf i. We shall determine the assignment HK R (q) := dim k (S/hX ^q , Y ^q , Z ^q , f i) .

Let f + X ^[q] be the ideal of S generated by all X ^q , Y ^q , Z ^q , and f . Let τ be the lexicographic order on S induced by the variable order X > Y > Z. Denote by in f + X ^[q]
the initial ideal of f + X ^[q] . Then by (2.11), we get that HK R (q) is equal to dim K S/in f + X ^[q]

.

By making use of Gr¨obner basis, we understand which monomials one has to add to fill the gap between in f + X ^[q]
and the ideal in(f) + X ^[q] .

We denote by [y] the greatest integer less than or equal to y, and S k (a, b) the element symmetric polynomial of a and b of degree k.

Let u = max{b, e}. We have

 q − 1 u



= min  q − 1 b



,  q − 1 e



for q ≫ 0. Define (v) ₊ = max{0, v}.

Let l u be the integer  _q−1

u , and ǫ be the remainder of q −1 divided by u. Then l u = ^q−1−ǫ _u and one has q − l u b > 0 and q − l u e > 0. On the other hand, at least one of (q − (l u + 1)b) ₊ and (q − (l u + 1)e) ₊ must be zero.

4.1 The case for a < d < e

Proposition 4.1 The Hilbert-Kunz function of the hypersurface X ^a Y ^b + Y ^c Z ^d + Z ^e

with 0 < a ≤ b ≤ c and a < 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

#

and f k (n) is an eventually periodic function of n for each k.

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Proof. Let τ be the lexicographic order on S induced by the variable order X > Y > Z.

Then X ^a Y ^b is bigger than Y ^c Z ^d and Z ^e .

First, we determine a Gr¨obner basis of the ideal

I q = hX ^q , Y ^q , Z ^q , f i, by the means of Buchberger’s algorithm (Algorithm 2.7).

By this algorithm, the elements

X ^q , Y ^q , Z ^q , X ^a Y ^b + Y ^c Z ^d + Z ^e , Y ^(q−ib)

Z ^ie , i = 1, . . . , γ − 1, X ^q−ja Z ^je − j

1



X ^q−(j+1)a Y ^c−b Z ^d+je + · · · +(−1) ^γ−j−1

 γ − 2 γ − j − 1



X ^{q−(γ−1)a} Y (γ−j−1)(c−b) Z (γ−j−1)d+je , j = 1, . . . , γ − 1,

, where γ = [ ^q−1 _a ], form a Gr¨obner basis of the ideal I q for q ≫ 0.

Thus the ideal in(I q ) is generated by

X ^q , Y ^q , Z ^q , X ^a Y ^b , Y ^(q−ib)

Z ^ie , i = 1, . . . , γ − 1, X ^q−ja Z ^je , j = 1, . . . , γ − 1.

That is, the ideal in(I q ) is generated by

X ^q , Y ^q , Z ^q , X ^a Y ^b , Y ^(q−δb)

Z ^δe , X ^(q−δa)

Z ^δe , δ = 1, . . . , l, where l = [ ^q−1 _e ], which is the same as the result in Proposition 3.1.

With the same method, we have the same conclusion of Theorem 3.4.

4.2 The case for d < e < a

Proposition 4.2 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 < a is

n 7−→ λp ²ⁿ + f 1 (n)p ⁿ + f 0 (n)

for n ≫ 0, where λ is a rational number and f k (n) is an eventually periodic function of n for each k.

Proof. Let τ be the lexicographic order on S induced by the variable order X > Y > Z.

Then X ^a Y ^b is bigger than Y ^c Z ^d and Z ^e .

First, we determine a Gr¨obner basis of the ideal

I q = hX ^q , Y ^q , Z ^q , f i,

12

by the means of Buchberger’s algorithm (Algorithm 2.7).

By this algorithm, the elements

X ^q , Y ^q , Z ^q , X ^a Y ^b + Y ^c Z ^d + Z ^e , Y ^(q−ib)

Z ^ie , i = 1, . . . , γ, X ^q−ja Z ^je − j

1



X ^q−(j+1)a Y ^c−b Z ^d+je + · · · +(−1) ^γ−j−1

 γ − 2 γ − j − 1



X ^{q−(γ−1)a} Y (γ−j−1)(c−b) Z (γ−j−1)d+je

+(−1) ^γ−j X ^q−γa Y ^{γc−(γ−j)b} Z ^γd + (−1) ^γ−j γ 1



X ^q−γa Y (γ−1)c−(γ−j)b Z ^(γ−1)d+e + · · · + (−1) ^γ−j

 γ j − 1



X ^q−γa Y (γ−j+1)c−(γ−j)b Z (γ−j+1)d+(j−1)e , j = 1, . . . , γ − 1,

, where γ = [ ^q−1 _a ], form a Gr¨obner basis of the ideal I q . Thus the ideal in(I q ) is generated by

X ^q , Y ^q , Z ^q , X ^a Y ^b , Y ^(q−ib)

Z ^ie , i = 1, . . . , γ, X ^q−ja Z ^je , j = 1, . . . , γ − 1.

That is, the ideal in(I q ) is generated by

X ^q , Y ^q , Z ^q , X ^a Y ^b , Y ^(q−δb)

Z ^δe , X ^(q−δa)

Z ^δe , δ = 1, . . . , l, where l = [ ^q−1 _e ], which is the same as the result in Proposition 3.1.

With the same method, we have the same conclusion of Theorem 3.4.

4.3 The case for d < a < e < b

Proposition 4.3 The Hilbert-Kunz function of the hypersurface X ^a Y ^b + Y ^c Z ^d + Z ^e

with 0 < a ≤ b ≤ c and d < a < e < b is

n 7−→ λp ²ⁿ + f ₁ (n)p ⁿ + f ₀ (n)

for n ≫ 0, where λ is a rational number and f k (n) is an eventually periodic function of n for each k.

Proof. Let τ be the lexicographic order on S induced by the variable order X > Y > Z.

Then X ^a Y ^b is bigger than Y ^c Z ^d and Z ^e .

First, we determine a Gr¨obner basis of the ideal

I q = hX ^q , Y ^q , Z ^q , f i,

13

by the means of Buchberger’s algorithm (Algorithm 2.7).

By this algorithm, the elements

X ^q , Y ^q , Z ^q , X ^a Y ^b + Y ^c Z ^d + Z ^e , Y ^(q−ib)

Z ^ie , i = 1, . . . , γ − 1, X ^q−ja Z ^je − j

1



X ^q−(j+1)a Y ^c−b Z ^d+je + · · · +(−1) ^γ−j−1

 γ − 2 γ − j − 1



X ^{q−(γ−1)a} Y (γ−j−1)(c−b) Z (γ−j−1)d+je

+(−1) ^γ−j X ^q−γa Y ^{γc−(γ−j)b} Z ^γd + (−1) ^γ−j γ 1



X ^q−γa Y (γ−1)c−(γ−j)b Z ^(γ−1)d+e + · · · + (−1) ^γ−j

 γ j − 1



X ^q−γa Y (γ−j+1)c−(γ−j)b Z (γ−j+1)d+(j−1)e , j = 1, . . . , γ − 1,

, where γ = [ ^q−1 _a ], form a Gr¨obner basis of the ideal I q . Thus the ideal in(I q ) is generated by

X ^q , Y ^q , Z ^q , X ^a Y ^b , Y ^(q−ib)

Z ^ie , i = 1, . . . , γ, X ^q−ja Z ^je , j = 1, . . . , γ − 1.

That is, the ideal in(I q ) is generated by

X ^q , Y ^q , Z ^q , X ^a Y ^b , Y ^(q−δb)

Z ^δe , X ^(q−δa)

Z ^δe , δ = 1, . . . , l, where l = [ ^q−1 _e ], which is the same as the result in Proposition 3.1.

With the same method, we have the same conclusion of Theorem 3.4.

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