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

2 The First Form : X^aY^b + Y ^cZ^d

Let K be a field of characteristic p > 0 and

S = K [X₁, . . . , X_r, Y₁, . . . , Y_s, Z₁, . . . , Z_t] .

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

f := X^aY^b+ Y^cZ^d

where X^a = X₁^a1. . . X_r^ar, Y^b = Y₁^b1. . . Y_s^bs, Y^c = Y₁^c1. . . Y_s^cs, Z^d = Z₁^d1. . . Z_t^dt, and r ≥ 1. Let q = pⁿ, J = j | bj > cj , and set R = S/ < f >. Then 0 ≤ |J| := m ≤ s, and w.l.o.g., we assume that b1 > c1, . . . , bm > cm, bm+1 ≤ cm+1, . . . , bs≤ cs. We shall determine the assignment

HKR(q) := dimK

 S < X₁^q, . . . , X_r^q, Y₁^q, . . . , Y_s^q, Z₁^q, . . . , Z_t^q, f >

 .

Let f + X^[q] be the ideal of S generated by all X_i^q’s, Y_j^q’s, Z_k^q’s, and f . Fix a term order on S, and denote by in f + X^[q]
the initial ideal of f + X^[q]. Then by Lemma 1.1, we get that HKR(q) is equal to dimK



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]. Throughout this section, it is not restrictive to assume that a₁ ≥ a₂ ≥ · · · ≥ a_r > 0 , b₁− c₁ ≥ b₂− c₂ ≥ · · · ≥ b_m − c_m > 0 , c_m+1− b_m+1 ≥ c_m+2 − b_m+2 ≥ · · · ≥ c_s− b_s ≥ 0, and d₁ ≥ d₂ ≥ · · · ≥ d_t> 0. Let u be the maximum of the integers a₁ , b₁− c₁ , c_m+1 − b_m+1, and d₁; that is, u is the greatest integer among all a_i’s, (b_j − c_j)’s, (c_h − b_h)’s, and d_k’s. We also denote by [y] the greatest integer less than or equal to y, and S_in(x) the elementary symmetric polynomial of degree i in n indeterminates x = (x₁, . . . , x_n). Let I_q be the ideal f + X^[q] and define (v)₊= max {0, v}.

In order to make it more easy to determine the Hilbert-Kunz function of R, we shall prove the following lemma.

Lemma 2.1. Let S = K [X₁, . . . , X_r, Y₁, . . . , Y_s], r ≥ 1, s ≥ 1, and the a_i, b_j, and c_j are all positive integers with b₁ − c₁ ≥ b₂ − c₂ ≥ · · · ≥ b_m − c_m > 0, c_m+1− b_m+1 ≥ c_m+2 − b_m+2 ≥

· · · ≥ c_s− b_s≥ 0. We denote by G the ideal generated by

X₁^q, . . . , X_r^q, Y₁^q, . . . , Y_s^q, X₁^[q−αa1]+Y^e+α(c−b)+, . . . , Xr^[q−αar]+Y^e+α(c−b)+, Y₁^{[q−α(b1−c1)−c1]+}Y^e+α(c−b)+, . . . , Ym^{[q−α(bm−cm)−cm]+}Y^e+α(c−b)+, and X^aY^b,

where α is a positive integer, e = (e₁, . . . , e_s), e₁ = c₁, . . . , e_m = c_m, e_m+1 = b_m+1, . . . , e_s = b_s, (c − b)₊= (0, . . . , 0, c_m+1− b_m+1, . . . , c_s− b_s). Then the dimension of S/G is equal to

q^r+s−

r

Y

i=1

(q − ai)

s

Y

j=1

(q − bj) − q^r

m

Y

j=1

(q − cj)

s

Y

h=m+1

[q − α(ch− b_h) − bh]₊

+

r

Y

i=1

(q − a_i)

m

Y

j=1

(q − b_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

+

r

Y

i=1

(q − αa_i)₊

m

Y

j=1

[q − α(b_j− c_j) − c_j ]₊

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

−

r

Y

i=1

[q − (α + 1)a_i ]₊

m

Y

j=1

[q − (α + 1)(b_j− c_j) − c_j ]₊

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊.

Proof : If [q − αa_i]₊ = 0 for some i or [q − α(b_j − c_j) − c_j ]₊ = 0 for some j, then G is generated by

X₁^q, . . . , X_r^q, Y₁^q, . . . , Y_s^q, Y^e+α(c−b)+, and X^aY^b. Hence,

dimK S/G

= q^r+s−

r

Y

i=1

(q − a_i)

s

Y

j=1

(q − b_j) − q^r

m

Y

j=1

(q − c_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

+

r

Y

i=1

(q − a_i)

m

Y

j=1

(q − b_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊ .

From now on, we assume q − αai > 0 for each i and q − α(bj− cj) − cj > 0 for each j. Let lα be the minimum of

  q − α(b_j− cj) − 1 c_j

 ,

 q − 1

b_h+ α(c_h− b_h)

  

 j = 1, . . . , m, h = m + 1, . . . , s

 .

Then we have q − α(b_j0 − c_j0) ≤ (l_α+ 1)c_j0 for some j₀ with 1 ≤ j₀ ≤ m or

q ≤ (lα+ 1)(bh0 + α(ch0 − bh0)) for some h0 with m + 1 ≤ h0 ≤ s, and q − α(b_j− c_j) − l_αc_j ≥ 1 for each j and q − l_α(b_h+ α(c_h− b_h)) ≥ 1 for each h.

We consider the ideals G_β = G : Y^β[^e+α(c−b)+], for β = 0, 1, 2, . . . , l_α + 1. Since G₀ = G, G_lα+1 = S, and G_β+1 = G_β : Y^e+α(c−b)+, we have the exact sequence of K-modules :

0 −→ S/G_β+1 ^{Ye+α(c−b)+}−→ S/G_β −→ S / < G_β, Y^e+α(c−b)+ > −→ 0.

It follows that

dim_K S/G
= dim_K S/G₀
=

lα

X

β=0

dim_K



S / < G_β, Y^e+α(c−b)+ >

 .

We determine dim_K

S / < G_β, Y^e+α(c−b)+ >

as follows : For β = 0, the ideal < G₀, Y^e+α(c−b)+ > is generated by

X₁^q, . . . , X_r^q, Y₁^q, . . . , Y_s^q, Y^e+α(c−b)+, and X^aY^b. Hence,

dim_K

S / < G₀, Y^e+α(c−b)+ >

= q^r+s−

r

Y

i=1

(q − a_i)

s

Y

j=1

(q − b_j) − q^r

m

Y

j=1

(q − c_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

+

r

Y

i=1

(q − a_i)

m

Y

j=1

(q − b_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊ .

For 1 ≤ β ≤ l_α, the ideal G_β = G : Y^β[^e+α(c−b)+] is generated by

X₁^q−αa1, . . . , X_r^q−αar, Y₁^{q−α(b1−c1)−βc1}, . . . , Ym^{q−α(bm−cm)−βcm}, Y_m+1^{q−β[bm+1+α(cm+1−bm+1)]}, . . . , Ys^{q−β[bs+α(cs−bs)]}, and X^aY₁^(b1−βc1)+· · · Ym^(bm−βcm)+Y_m+1^{[bm+1−β(bm+1+α(cm+1−bm+1))]+}· · · Ys^{[bs−β(bs+α(cs−bs))]+} .

Hence, the ideal < Gβ, Y^e+α(c−b)+ > is generated by

X₁^q−αa1, . . . , X_r^q−αar, Y₁^{q−α(b1−c1)−βc1}, . . . , Ym^{q−α(bm−cm)−βcm}, Y_m+1^{q−β[bm+1+α(cm+1−bm+1)]}, . . . , Ys^{q−β[bs+α(cs−bs)]}, X^aY₁^(b1−βc1)+· · · Ym^(bm−βcm)+Y_m+1^{[bm+1−β(bm+1+α(cm+1−bm+1))]+}· · ·

Ys^{[bs−β(bs+α(cs−bs))]+}, and Y^e+α(c−b)+ .

Thus, dim_K

S / < G_β, Y^e+α(c−b)+ >

=

r

Y

i=1

(q − αa_i)

m

Y

j=1

[q − α(b_j − c_j) − βc_j ]₊

s

Y

h=m+1

[q − β(b_h+ α(c_h− b_h))]₊−

r

Y

i=1

[q − (α + 1)ai]₊

m

Y

j=1

q − α(bj − cj) − βcj − (bj − βcj)₊

+ s

Y

h=m+1

[q − β(bh+ α(ch− bh))]₊

−

r

Y

i=1

(q − αa_i)

m

Y

j=1

[q − α(b_j− c_j) − (β + 1)c_j ]₊

s

Y

h=m+1

[q − (β + 1)(b_h+ α(c_h− b_h))]₊

+

r

Y

i=1

[q − (α + 1)a_i]₊

m

Y

j=1

[q − α(b_j − c_j) − βc_j − u_jβ]₊

s

Y

h=m+1

[q − (β + 1)(b_h+ α(c_h− b_h))]₊

=

r

Y

i=1

(q − αa_i) × ( _m

Y

j=1

[q − α(b_j − c_j) − βc_j ]₊

s

Y

h=m+1

[q − β (b_h+ α (c_h − b_h))]₊

−

m

Y

j=1

[q − α(b_j− c_j) − (β + 1)c_j ]₊

s

Y

h=m+1

[q − (β + 1)(b_h+ α(c_h− b_h))]₊ )

−

r

Y

i=1

[q − (α + 1)ai]₊× ( _m

Y

j=1

q − α(bj− cj) − βcj− (bj − βcj)₊

+ s

Y

h=m+1

[q − β(bh + α(ch−

b_h))]₊−

m

Y

j=1

[q − α(b_j− c_j) − βc_j− u_jβ]₊

s

Y

h=m+1

[q − (β + 1)(b_h+ α(c_h− b_h))]₊ )

,

where u_jβ = max cj , [b_j− βc_j ]₊ . Now, we have

dimK S/G

= q^r+s−

r

Y

i=1

(q − a_i)

s

Y

j=1

(q − b_j) − q^r

m

Y

j=1

(q − c_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

+

r

Y

i=1

(q − a_i)

m

Y

j=1

(q − b_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

+

r

Y

i=1

(q − αa_i) × ( _l

α

X

β=1

" _m Y

j=1

[q − α(b_j − c_j) − βc_j ]₊

s

Y

h=m+1

[q − β(b_h+ α(c_h− b_h))]₊

−

m

Y

j=1

[q − α(b_j− c_j) − (β + 1)c_j ]₊

s

Y

h=m+1

[q − (β + 1)(b_h+ α(c_h− b_h))]₊

#)

−

r

Y

i=1

[q − (α + 1)a_i]₊ × ( _lα

X

β=1

" _m Y

j=1

[q − α(b_j − c_j) − βc_j− (b_j− βc_j)₊]₊

s

Y

h=m+1

[q − β(b_h+

α(ch− bh))]₊−

m

Y

j=1

[q − α(bj − cj) − βcj − ujβ]₊

s

Y

h=m+1

[q − (β + 1)(bh+ α(ch− bh))]₊

#) .

Let (∗) be the term

lα

X

β=1

" _m Y

j=1

[q − α(b_j− c_j) − βc_j ]₊

s

Y

h=m+1

[q − β(b_h+ α(c_h− b_h))]₊

−

m

Y

j=1

[q − α(b_j − c_j) − (β + 1)c_j ]₊

s

Y

h=m+1

[q − (β + 1)(b_h+ α(c_h− b_h))]₊

# . Since q − α(b_j0 − c_j0) ≤ (l_α+ 1)c_j0 for some j₀ with 1 ≤ j₀ ≤ m or

q ≤ (lα+ 1) bh0+ α(ch0 − bh0)

for some h0 with m + 1 ≤ h0 ≤ s, (∗) is equal to

m

Y

j=1

[q − α(b_j− c_j) − c_j ]₊

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊.

Let (∗∗) be the term

lα

X

β=1

" _m Y

j=1

q − α(b_j− c_j) − βc_j− (b_j − βc_j)₊

+ s

Y

h=m+1

[q − β(b_h+ α(c_h− b_h))]₊

−

m

Y

j=1

[q − α(b_j − c_j) − βc_j − u_jβ ]₊

s

Y

h=m+1

[q − (β + 1)(b_h+ α(c_h− b_h))]₊

# . Since

m

Y

j=1

[q − α(b_j− c_j) − βc_j− u_jβ ]₊

s

Y

h=m+1

[q − (β + 1)(b_h+ α(c_h− b_h))]₊

=

m

Y

j=1

q − α(b_j − c_j) − (β + 1)c_j − (b_j− (β + 1)c_j)₊

+ s

Y

h=m+1

[q − (β + 1)(b_h+ α(c_h− b_h))]₊,

where β = 1, 2, . . . , l_α− 1, the term (∗∗) is equal to

m

Y

j=1

[q − (α + 1)(b_j − c_j) − c_j ]₊

s

Y

h=m+1

[q − α(c_h − b_h) − b_h]₊

−

m

Y

j=1

[q − α(b_j − c_j) − l_αc_j − u_jlα ]₊

s

Y

h=m+1

[q − (l_α+ 1)(b_h+ α(c_h− b_h))]₊.

Since q − α(b_j0 − c_j0) ≤ (l_α+ 1)c_j0 or q ≤ (l_α+ 1) b_h0+ α(c_h0 − b_h0)
, we have

m

Y

j=1

[q − α(b_j − c_j) − l_αc_j− u_jlα ]₊ = 0 or

s

Y

h=m+1

[q − (l_α+ 1)(b_h+ α(c_h− b_h))]₊= 0 . Thus, (∗∗) is equal to

m

Y

j=1

[q − (α + 1)(b_j− c_j) − c_j ]₊

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊.

So,

dim_K S/G

= q^r+s−

r

Y

i=1

(q − a_i)

s

Y

j=1

(q − b_j) − q^r

m

Y

j=1

(q − c_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

+

r

Y

i=1

(q − a_i)

m

Y

j=1

(q − b_j)

s

Y

h=m+1

[q − α(c_h − b_h) − b_h]₊

+

r

Y

i=1

(q − αai)₊

m

Y

j=1

[q − α(bj− cj) − cj ]₊

s

Y

h=m+1

[q − α(ch− bh) − bh]₊

−

r

Y

i=1

[q − (α + 1) a_i ]₊

m

Y

j=1

[q − (α + 1)(b_j− c_j) − c_j ]₊

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊.



Since u is the maximum of the integers among all a_i’s, (b_j − c_j)’s, (c_h− b_h)’s, and d_k’s, we have

_q−v

u  := min (

hq−1 ai

i ,h_q−c

j−1 bj−c_j

i ,h

q−bh−1 ch−b_h

i ,h

q−1 dk

i   

1 ≤ i ≤ r, b_j − c_j > 0, c_h− b_h > 0 1 ≤ j ≤ m, m + 1 ≤ h ≤ s, 1 ≤ k ≤ t

)

for q  0, where v = 1 or 1 + c_j for some j or 1 + b_h for some h.

Let l_u be the integer_q−v

u , and  be the remainder of q − v divided by u. Then l_u = ^q−v−_u and one has q − l_ua_i > 0, q − l_u(b_j− c_j) − c_j > 0, q − l_u(c_h− b_h) − b_h > 0, and q − l_ud_k > 0 for all i, j, h, and k. On the other hand, by the definition of l_u, at least one of [q − (l_u+ 1)a_i]₊’s, [q − (l_u+ 1)(b_j − c_j) − c_j ]₊’s, [q − (l_u+ 1)(c_h− b_h) − b_h]₊’s, and [q − (l_u+ 1)d_k]₊’s must be zero.

Proposition 2.2. Let f := X^aY^b+ Y^cZ^d. Then HK_R(q) = q^r+s+t− q^t

r

Y

i=1

(q − a_i)

s

Y

j=1

(q − b_j)

− q^r

m

Y

j=1

(q − c_j)×

( _l

u

X

α=1

" _s Y

h=m+1

[q − α(c_h− b_h) − b_h]

#

×

" _t Y

k=1

(q − αd_k) −

t

Y

k=1

[q − (α + 1)d_k]₊

#)

+

r

Y

i=1

(q − a_i)

m

Y

j=1

(q − b_j)×

( _l

u

X

α=1

" _s Y

h=m+1

[q − α(c_h− b_h) − b_h]

#

×

" _t Y

k=1

(q − αd_k) −

t

Y

k=1

[q − (α + 1)d_k]₊

#)

+

lu

X

α=1

( _r Y

i=1

(q − αa_i)

m

Y

j=1

[q − α(b_j − c_j) − c_j ]

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]

−

r

Y

i=1

[q − (α + 1)a_i]₊

m

Y

j=1

[q − (α + 1)(b_j− c_j) − c_j]₊

s

Y

h=m+1

[q − α(c_h− b_h) − b_h] )

× ( _t

Y

k=1

(q − αdk) −

t

Y

k=1

[q − (α + 1)dk]₊ )

, where l_u is the integer  _q−v

u , and 0 ≤ m ≤ s .

Proof : Let u be the maximum of the integers a₁, b₁− c₁, c_m+1− b_m+1, and d₁. Let < be the lexicographic order on S and define

e_j = c_j for j = 1, . . . , m and e_h = b_h for h = m + 1, . . . , s.

Then X^aY^b is bigger than Y^cZ^d and Y^e = Y₁^e1. . . Y_s^es = Y₁^c1. . . Y_m^cmY_m+1^bm+1. . . Y_s^bs. We determine a Gr¨obner basis of the ideal

I_q =< X₁^q, . . . , X_r^q, Y₁^q, . . . , Y_s^q, Z₁^q, . . . , Z_t^q, f >,

by means of Buchberger’s algorithm (Algorithm 1.9). By this algorithm, the elements X₁^q, . . . , X_r^q, Y₁^q, . . . , Y_s^q, Z₁^q, . . . , Z_t^q, X_i^(q−δai)+Y^e+δ(c−b)+Z^δd, i = 1, . . . , r, δ = 1, . . . , l, Y_j^{[q−δ(bj−cj)−cj]+}Y^e+δ(c−b)+Z^δd, j = 1, . . . , m, δ = 1, . . . , l, and X^aY^b+ Y^cZ^d,

form a Gr¨obner basis of the ideal I_q, where l =h

q−1 d1

i

. Thus, the ideal in(I_q) is generated by X₁^q, . . . , X_r^q, Y₁^q, . . . , Y_s^q, Z₁^q, . . . , Z_t^q, X_i^(q−δai)+Y^e+δ(c−b)+Z^δd, i = 1, . . . , r, δ = 1, . . . , l, Y_j^{[q−δ(bj−cj)−cj]+}Y^e+δ(c−b)+Z^δd, j = 1, . . . , m, δ = 1, . . . , l, and X^aY^b.

Now we have to compute the dimension of S/in(I_q). In order to do this, we consider the ideals K_α = in(I_q) : Z^αd for α = 0, 1, . . . , l + 1, where Z^αd = Z₁^αd1. . . Z_t^αdt. Since K₀ = in(I_q), K_l+1 = S, and K_α+1 = K_α : Z^d, we have the exact sequence of K-modules :

0 −→ S/K_α+1 −→ S/K^Zd _α −→ S / < K_α, Z^d> −→ 0 . It follows that

dim_K S/in(I_q)
= dim_K S/K₀
=

l

X

α=0

dim_K

S / < K_α, Z^d> .

We compute dim_K

S / < K_α, Z^d>

as follows : For α = 0, the ideal < K₀, Z^d> is generated by

X₁^q, . . . , X_r^q, Y₁^q, . . . , Y_s^q, Z₁^q, . . . , Z_t^q, X^aY^b, and Z^d. Let S₁ = K [X₁, . . . , X_r, Y₁, . . . , Y_s], and S₂ = K [Z₁, . . . , Z_t]. Then

dim_K

S / < K₀, Z^d>

= dim_K

S₁ / < X₁^q, . . . , X_r^q, Y₁^q, . . . , Y_s^q, X^aY^b >

× dim_K

S₂ / < Z₁^q, . . . , Z_t^q, Z^d> 

=

"

q^r+s−

r

Y

i=1

(q − a_i)

s

Y

j=1

(q − b_j)

#

×

"

q^t−

t

Y

k=1

(q − d_k)

#

= q^r+s+t− q^t

r

Y

i=1

(q − a_i)

s

Y

j=1

(q − b_j) − q^r+s

t

Y

k=1

(q − d_k) +

r

Y

i=1

(q − a_i)

s

Y

j=1

(q − b_j)

t

Y

k=1

(q − d_k).

For 1 ≤ α ≤ l, the ideal K_α = in(I_q) : Z^αd is generated by

X₁^q, . . . , X_r^q, Y₁^q, . . . , Y_s^q, Z₁^q−αd1, . . . , Z_t^q−αdt, X_i^(q−δai)+Y^e+δ(c−b)+Z^(δ−α)d, i = 1, . . . , r, δ = 1, . . . , l, Y_j^{[q−δ(bj−cj)−cj]+}Y^e+δ(c−b)+Z^(δ−α)d, j = 1, . . . , m, δ = 1, . . . , l, and X^aY^b.

Hence, the ideal < K_α, Z^d> is generated by

X₁^q, . . . , X_r^q, Y₁^q, . . . , Y_s^q, Z₁^q−αd1, . . . , Z_t^q−αdt, X₁^(q−αa1)+Y^e+α(c−b)+, . . . , Xr^(q−αar)+Y^e+α(c−b)+, Y₁^{[q−α(b1−c1)−c1]+}Y^e+α(c−b)+, . . . , Ym^{[q−α(bm−cm)−cm]+}Y^e+α(c−b)+, X^aY^b, and Z^d.