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

3 The Second Form : Y ^b + Y ^cZ^d

Again, let K be a field of characteristic p > 0 and

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

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

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

where Y^b = Y₁^b1. . . Y_s^bs, Y^c = Y₁^c1. . . Y_s^cs, Z^d = Z₁^d1. . . Z_t^dt. 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  < Y₁^q, . . . , Y_s^q, Z₁^q, . . . , Z_t^q, f >

 .

Throughout this section, it is not restrictive to assume that b1 − c1 ≥ b2 − c2 ≥ · · · ≥ bm− cm > 0 , cm+1− bm+1 ≥ cm+2 − bm+2 ≥ · · · ≥ cs− bs ≥ 0, and d1 ≥ d2 ≥ · · · ≥ dt > 0.

Let u be the maximum of the integers b1− c1, cm+1− bm+1, and d1; that is, u is the greatest integer among all (bj− cj)’s , (ch− bh)’s, and dk’s. We also denote by [y] the greatest integer less than or equal to y, and Sin(x) the elementary symmetric polynomial of degree i in n indeterminates x = (x1, . . . , xn). Let Iq be the ideal of S generated by all Y_j^q’s, Z_k^q’s, and f , and define (v)₊ = max {0, v}.

Firstly, we prove the following lemma.

Lemma 3.1. Let S = K [Y₁, . . . , Y_s], s ≥ 1, and G the ideal generated by

Y₁^q, . . . , Y_s^q, Y₁^{[q−α(b1−c1)−c1]+}Y^e+α(c−b)+, . . . , Ym^{[q−α(bm−cm)−cm]+}Y^e+α(c−b)+, and Y^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, and (c − b)₊ = (0, . . . , 0, c_m+1 − b_m+1, . . . , c_s− b_s). Then the dimension of S/G is equal to

q^s−

s

Y

j=1

(q − b_j) −

m

Y

j=1

(q − c_j)

s

Y

h=m+1

[q − α(c_h − b_h) − b_h]₊

+

m

Y

j=1

(q − b_j)

s

Y

h=m+1

[q − α(c_h − b_h) − b_h]₊

+

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]₊

−

m

Y

j=1

[q − (α + 1)(bj − cj) − cj]₊

s

Y

h=m+1

[q − α(ch− bh) − bh]₊.

Proof : If [q − α(b_j − c_j) − c_j ]₊ = 0 for some j with 1 ≤ j ≤ m, then G is generated by Y₁^q, . . . , Y_s^q, Y^e+α(c−b)+, and Y^b.

Thus,

dim_K S/G
= q^s−

s

Y

j=1

(q − b_j) −

m

Y

j=1

(q − c_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

+

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 − α(b_j − c_j) − c_j > 0 for each j = 1, 2, . . . , m. Let l_α be the minimum of

  q − α(b_j− c_j) − 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 ≤ (l_α+ 1) b_h0+ α(c_h0− b_h0)

for some h₀ with m + 1 ≤ h₀ ≤ s, and q − α(bj− cj) − lαcj ≥ 1 for each j and q − lα(bh+ α(ch− bh)) ≥ 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 compute 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 Y₁^q, . . . , Y_s^q, Y^e+α(c−b)+, and Y^b. Thus,

dim_K

S / < G₀, Y^e+α(c−b)+ >

= q^s−

s

Y

j=1

(q − b_j) −

m

Y

j=1

(q − c_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

+

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

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 Y₁^(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))]+} .

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

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)]},

Y₁^(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)+. Hence,

dim_K

S / < G_β, Y^e+α(c−b)+ >

=

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))]₊

−

m

Y

j=1

q − α(bj − c_j) − βc_j− (b_j− βc_j)₊

+ s

Y

h=m+1

[q − β(b_h+ α(c_h− b_h)) − [b_h− 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))]₊,

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

dim_K S/G
= q^s−

s

Y

j=1

(q − b_j) −

m

Y

j=1

(q − c_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

+

m

Y

j=1

(q − bj)

s

Y

h=m+1

[q − α(ch− bh) − bh]₊

+

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(bj − cj) − (β + 1)cj ]₊

s

Y

h=m+1

[q − (β + 1)(bh+ α(ch− bh))]₊ )

−

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(bj − cj) − βcj − ujβ]₊

s

Y

h=m+1

[q − (β + 1)(bh+ α(ch− bh))]₊ )

.

Let (4) 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,

the term (4) 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 (44) 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(bj − 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 (44) 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(bj− cj) − lαcj− ujlα ]₊

s

Y

h=m+1

[q − (lα+ 1)(bh+ α(ch− bh))]₊.

Since q − α(bj0 − cj0) ≤ (lα+ 1)cj0 or q ≤ (lα+ 1) bh0 + α(ch0 − bh0)
, we have

m

Y

j=1

[q − α(bj − cj) − lαcj− ujlα ]₊ = 0 or

s

Y

h=m+1

[q − (lα+ 1)(bh+ α(ch − bh))]₊ = 0.

Thus, (44) is equal to

m

Y

j=1

[q − (α + 1)(bj− cj) − cj ]₊

s

Y

h=m+1

[q − α(ch− bh) − bh]₊.

Therefore,

dimK S/G
= q^s−

s

Y

j=1

(q − bj) −

m

Y

j=1

(q − cj)

s

Y

h=m+1

[q − α(ch− bh) − bh]₊

+

m

Y

j=1

(q − b_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

+

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]₊

−

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 (b_j − c_j)’s, (c_h− b_h)’s, and d_k’s, we have

 _q−v

u  := min (

h _q−c

j−1 bj−cj

i ,h

q−b_h−1 ch−b_h

i ,h

q−1 dk

i   

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 + cj for some j or 1 + bh for some h. Let lu 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_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 j, h, and k. On the other hand, by the definition of l_u, at least one of [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 3.2. Let f := Y^b+ Y^cZ^d. Then HK_R(q) is equal to

q^s+t− q^t

s

Y

j=1

(q − bj)

−

m

Y

j=1

(q − c_j) × ( _lu

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]₊

#)

+

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

( _m Y

j=1

[q − α(bj − cj) − cj ]

s

Y

h=m+1

[q − α(ch− bh) − bh] −

m

Y

j=1

[q − (α + 1)(bj− cj) − cj ]₊

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]₊ )

,

where l_u is the integer  _q−v

u , and 0 ≤ m ≤ s .

Proof : We prove this proposition by discussing on m, and let < be the lexicographic order on S.

Case 1 : Assume that m = 0, i.e., b_j ≤ c_j for each j = 1, 2, . . . , s. Then Y^cZ^d is the leading term of f . The elements

Y₁^q, . . . , Y_s^q, Z₁^q, . . . , Z_t^q, and Y^b

form a Gr¨obner basis of the ideal I_q . Thus, the ideal in(I_q) is generated by the elements as above. It follows that

dim_K S/in (I_q)
= dim_K

S / < Y₁^q, . . . , Y_s^q, Z₁^q, . . . , Z_t^q, Y^b > .

Hence, HK_R(q) = q^s+t− q^tQs

j=1(q − b_j) . Case 2 : Suppose that 1 ≤ m ≤ s, and define

ej = cj for 1 ≤ j ≤ m, and eh = bh for m + 1 ≤ h ≤ s.

Then Y^b is the leading term of f and Y^e = Y₁^e1· · · Y_s^es = Y₁^c1· · · Y_m^cmY_m+1^bm+1· · · Y_s^bs . By means of Buchberger’s algorithm (Algorithm 1.9), the elements

Y₁^q, . . . , Y_s^q, Z₁^q, . . . , Z_t^q, Y^b+ Y^cZ^d, and

Y_j^{[q−δ(bj−cj)−cj]+}Y^e+δ(c−b)+Z^δd, j = 1, . . . , m, δ = 1, . . . , l, form a Gr¨obner basis of the ideal I_q, where l =h

q−1 d1

i . Hence, the ideal in(I_q) is generated by

Y₁^q, . . . , Y_s^q, Z₁^q, . . . , Z_t^q, Y^b, and Y_j^{[q−δ(bj−cj)−cj]+}Y^e+δ(c−b)+Z^δd, j = 1, . . . , m, δ = 1, . . . , l.

Now we have to compute the dimension of S/in(Iq). In order to do this, we consider the ideals Kα = in(Iq) : Z^αd for α = 0, 1, . . . , l + 1. Since K0 = in(Iq), Kl+1 = S, and Kα+1 = Kα : Z^d, we have the exact sequence of K-modules :

0 −→ S/Kα+1 Z^d

−→ S/Kα −→ S / < Kα, Z^d> −→ 0.

It follows that

dim_K S/in(I_q)
= dimK 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 Y₁^q, . . . , Y_s^q, Z₁^q, . . . , Z_t^q, Y^b, and Z^d. Therefore,

dim_K

S / < K₀, Z^d>

= dim_K

S / < Y₁^q, . . . , Y_s^q, Z₁^q, . . . , Z_t^q, Y^b, Z^d >

= q^s+t− q^t

s

Y

j=1

(q − b_j) − q^s

t

Y

k=1

(q − d_k) +

s

Y

j=1

(q − b_j)

t

Y

k=1

(q − d_k).

For 1 ≤ α ≤ l, the ideal < K_α, Z^d> is generated by

Y₁^q, . . . , Y_s^q, Z₁^q−αd1, . . . , Z_t^q−αdt, Y^b, Z^d, and Y_j^{[q−α(bj−cj)−cj]+}Y^e+δ(c−b)+, j = 1, . . . , m.

Let S₁ = K [Y₁, . . . , Y_s], and S₂ = K [Z₁, . . . , Z_t]. Then by Lemma 3.1, we have dim_K

S / < K_α, Z^d>

= dim_K

S₁ / < Y₁^q, . . . , Y_s^q, Y^b, Y₁^{[q−α(b1−c1)−c1]+}Y^e+α(c−b)+, . . . , Ym^{[q−α(bm−cm)−cm]+}Y^e+α(c−b)+ >

× dim_K

S₂ / < Z₁^q−αd1, . . . , Z_t^q−αdt, Z^d>

= (

q^s−

s

Y

j=1

(q − b_j) −

m

Y

j=1

(q − c_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

+

m

Y

j=1

(q − b_j)

s

Y

h=m+1

[q − α(c_h− b_h) − b_h]₊

+

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]₊

−

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 − αd_k) −

t

Y

k=1

[q − (α + 1)d_k]₊ )

.

Since dim_K S/in(I_q)

can be written as

dim_K

S / < K₀, Z^d> +

l

X

α=1

dim_K

S / < K_α, Z^d > ,

it follows that dimK S/in(Iq)

is equal to q^s+t− q^t

s

Y

j=1

(q − b_j) − q^s

t

Y

k=1

(q − d_k) +

s

Y

j=1

(q − b_j)

t

Y

k=1

(q − d_k)

+

"

q^s−

s

Y

j=1

(q − bj)

#

× ( _l

X

α=1

" _t Y

k=1

(q − αdk) −

t

Y

k=1

[q − (α + 1)dk]₊

# )

−

l

X

α=1

( _m Y

j=1

(q − c_j)

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]₊ )

+

l

X

α=1

( _m Y

j=1

(q − b_j)

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]₊ )

+

l

X

α=1

( _m Y

j=1

[q − α(bj− cj) − cj]₊

s

Y

h=m+1

[q − α(ch− bh) − bh]₊−

m

Y

j=1

[q − (α + 1)(bj − cj) − cj]₊

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]₊ )

.

Let (444) be the term

"

q^s−

s

Y

j=1

(q − b_j)

#

× ( _l

X

α=1

" _t Y

k=1

(q − αd_k) −

t

Y

k=1

[q − (α + 1)d_k]₊

# ) .

Since l =h

q−1 d1

i

, we have q ≤ (l + 1)d₁, and so Qt

k=1[q − (α + 1)d_k]₊ = 0.

Thus, the term (444) is equal to

q^s

t

Y

k=1

(q − d_k) −

s

Y

j=1

(q − b_j)

t

Y

k=1

(q − d_k).

It follows that dim_K S/in(I_q)

is equal to

q^s+t− q^t

s

Y

j=1

(q − bj)

−

m

Y

j=1

(q − c_j) × ( _l

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]₊

#)

+

m

Y

j=1

(q − b_j) × ( _l

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]₊

#)

+

l

X

α=1

( _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]₊−

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 − αd_k) −

t

Y

k=1

[q − (α + 1)d_k]₊ )

.

By the definition of l_u, we have HK_R(q) = q^s+t− q^t

s

Y

j=1

(q − b_j)

−

m

Y

j=1

(q − c_j) × ( _lu

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]₊

#)

+

m

Y

j=1

(q − b_j) × ( _lu

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

( _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] −

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 − αd_k) −

t

Y

k=1

[q − (α + 1)d_k]₊ )

.