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In this section we shall determine the Hilbert-Kunz function of the hypersurfaces of the following form:

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3 The first case: d ≥ e

Let K be a field of characteristic p > 0 and

S = K[X, Y, Z].

In this section we shall determine 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 and d ≥ e. 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.

Proposition 3.1 Let f := X a Y b + Y c Z d + Z e with 0 < a ≤ b ≤ c and d ≥ e. Then

HK R (q) = (a + b)eq − abe +

l

u

X

α=1

[(q − αa)(q − αb) − (q − (α + 1)a) + (q − (α + 1)b) + ]

× [(q − αe − (q − (α + 1)e) + ] , where l u is the integer [ q−1 u ].

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,

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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−γa Z γe , 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, form a Gr¨obner basis of the ideal I q , where γ = [ q−1 a ].

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.

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 ].

Second, we compute the dimension of S/in(I q ).

Consider the ideals K α := (in(I q ) : Z αe ) for α = 0, 1, . . . , l + 1. Since K 0 = in(I q ), K l+1 = S, and K α+1 = (K α : Z e ), we get an exact sequence of K-modules:

0 −→ S/K α+1 −→ S/K α −→ S/hK α , Z e i −→ 0.

It follows that

dim K (S/in(I q )) = dim K (S/K 0 ) =

l

X

α=0

dim K (S/hK α , Z e i) .

We compute dim K (S/hK α , Z e i) as follows:

For α = 0, the ideal hK 0 , Z e i is generated by

X q , Y q , X a Y b , Z e . Then

dim K (S/hK 0 , Z e i) = e[q 2 − (q − a)(q − b)] = (a + b)eq − abe.

For 1 ≤ α ≤ l, the ideal K α := (in(I q ) : Z αe ) is generated by

X q , Y q , Z q−αe , X a Y b , Y q−δb Z (δ−α)e , X q−δa Z (δ−α)e , δ = 1, . . . , l.

Thus, the ideal hK α , Z e i is generated by

X q , Y q , Z q−αe , X a Y b , X (q−αa)

+

, Y (q−αb)

+

, Z e .

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If (q − δa) + = 0 or (q − δb) + = 0, then hK α , Z e i is generated by X q , Y q , Z q−αe , X a Y b , Z e . Hence,

dim K (S/hK α , Z e i) = [q 2 − (q − αa)(q − αb)][q − αe − (q − (α + 1)e) + ].

If q − δa > 0 and q − δb > 0, then hK α , Z e i is generated by X q , Y q , Z q−αe , X a Y b , X q−αa , Y q−αb , Z e . Therefore,

dim K (S/hK α , Z e i) = [(q−αa)(q−αb)−(q−(α+1)a) + (q−(α+1)b) + ][q−αe−(q−(α+1)e) + ].

Now, we have

dim K (S/in(I q )) = (a + b)eq − abe +

β

X

α=1

[(q − αa)(q − αb) − (q − (α + 1)a) + (q − (α + 1)b) + ]

× [(q − αe − (q − (α + 1)e) + ] +

l

X

α=β+1

[q 2 − (q − αa)(q − αb)][q − αe − (q − (α + 1)e) + ],

where β = [ q−1 b ].

By the definition of l u , we have HK R (q) = dim K (S/in(I q ))

= (a + b)eq − abe +

l

u

X

α=1

[(q − αa)(q − αb) − (q − (α + 1)a) + (q − (α + 1)b) + ]

× [(q − αe − (q − (α + 1)e) + ] .

In order to make it easier to observe the behavior of the Hilbert-Kunz function of R, we shall prove the next two lemmas.

Lemma 3.2

l

u

−1

X

α=1

[(q − αa)(q − αb) − (q − (α + 1)a)(q − (α + 1)b)] × [(q − αe − (q − (α + 1)e)]

=

" 2 X

k=1

(−1) k+1 S k (a, b) e u k

#

q 2 + ( terms of degree ≤ 1 in q over Q[ǫ] ),

where Q[ǫ] is the polynomial ring in ǫ over Q.

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Proof. Let A = (q − αa)(q − αb) = q 2 − α(a + b)q + α 2 ab,

B = (q − (α + 1)a)(q − (α + 1)b) = q 2 − (α + 1)(a + b)q + (α + 1) 2 ab, and C = [(q − αe − (q − (α + 1)e)] = e.

Then

A − B = [(α + 1) − α](a + b)q − (α + 1) 2 − α 2  ab.

Thus,

l

u

−1

X

α=1

A − B =

2

X

k=1

"

(−1) k+1 S k (a, b) · q 2−k ·

l

u

−1

X

α=1

W k (α)

# , where W k (α) = (α + 1) k − α k .

Since

W k (α) = (α + 1) k − α k = kα k−1 + k 2



α k−2 + · · · , it follows that

l

u

−1

X

α=1

W k (α) = k

l

u

−1

X

α=1

α k−1 + k 2

 l

u

−1

X

α=1

α k−2 + · · · ,

= l k u + ( terms of degree ≤ k − 1 in l u over Q ) . Replacing l u with q−1−ǫ u , we obtain the following expression:

l

u

−1

X

α=1

W k (α) =  q − 1 − ǫ u

 k

+



terms of degree ≤ k − 1 in q − 1 − ǫ

u over Q



= 1

u k q k + ( terms of degree ≤ k − 1 in q over Q[ǫ] ) , where Q[ǫ] is the polynomial ring in ǫ over Q.

So, we know

l

u

−1

X

α=1

(A − B)C =

" 2 X

k=1

(−1) k+1 S k (a, b) e u k

#

q 2 + ( terms of degree ≤ 1 in q over Q[ǫ] ).

Lemma 3.3

[(q − l u a)(q − l u b) − (q − (l u + 1)a) + (q − (l u + 1)b) + ] × [(q − l u e − (q − (l u + 1)e) + ] (∗)

= terms of degree ≤ 1 in q over Q[ǫ], for q ≫ 0.

Proof. We prove this lemma by discussing on u.

Case 1: Suppose u = e. Then (q − (l u + 1)e) + = 0, and l u = q−1−ǫ e . If e > b, then q − (l u + 1)a > 0 and q − (l u + 1)b > 0 for q > (e−1)b e−b . It follows that for q ≫ 0, (∗) is equal to

[(q − l u a)(q − l u b) − (q − (l u + 1)a)(q − (l u + 1)b)] × (q − l u e).

Replacing l u with q−1−ǫ e , we obtain the following expression:

1 + ǫ

e 2 {[(e − a)q + (1 + ǫ)a][(e − b)q + (1 + ǫ)b] − [(e − a)q + (1 + ǫ − e)a][(e − b)q + (1 + ǫ − e)b]}.

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Therefore, (∗) can be expressed as a polynomial in q of degree ≤ 1 over Q[ǫ], for q ≫ 0.

If e = b, then (q − (l u + 1)b) + = 0.

In this case, (∗) is equal to

(q − l u a)(q − l u b) × (q − l u e).

Replacing l u with q−1−ǫ e , we obtain the following expression:

(1 + ǫ) 2

e [(e − b)q + (1 + ǫ)b].

Hence, (∗) can be expressed as a polynomial in q of degree ≤ 1 over Q[ǫ], for q ≫ 0.

Case 2: Suppose u = b. Then (q − (l u + 1)b) + = 0, and l u = q−1−ǫ b . If b = e, see Case 1.

If b > e, then q − (l u + 1)e > 0 for q > (b−1)e b−e . It follows that for q ≫ 0, (∗) is equal to

e(q − l u a)(q − l u b).

Replacing l u with q−1−ǫ e , we obtain the following expression:

1 + ǫ

b 2 [(a − b)q + (1 + ǫ)b]e.

So, (∗) can be expressed as a polynomial in q of degree ≤ 1 over Q[ǫ], for q ≫ 0.

Theorem 3.4 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

#

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

Proof. Let q = p n . By Proposition 3.1, HK R (q) can be written as the sum of the following three parts:

(a + b)eq − abe, (1)

l

u

−1

X

α=1

[(q − αa)(q − αb) − (q − (α + 1)a)(q − (α + 1)b)] × [(q − αe − (q − (α + 1)e)] , (2) [(q − l u a)(q − l u b) − (q − (l u + 1)a) + (q − (l u + 1)b) + ] × [(q − l u e − (q − (l u + 1)e) + ] . (3) By applying Lemma 3.2 and Lemma 3.3 to the three parts, we obtain that

HK R (q) = λq 2 + ∆ 1 (ǫ)q + ∆ 0 (ǫ), where λ =

" 2 X

k=1

(−1) k+1 S k (a, b) e u k

#

, u = max{b, e}, and ∆ 1 (ǫ), ∆ 0 (ǫ) are polynomials in ǫ over Q.

Let f k (n) := ∆ k (ǫ), k = 0, 1. Since ǫ is the remainder of q − 1 divide by u, f k (n) is an

eventually periodic function of n for each n ≫ 0.

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