for *q*_{2}_{,}_{1}(*X, Y*)*, q*_{2}_{,}_{2}(*X, Y*) *∈* *K*[*X, Y*] with LM(*q*_{2}_{,}_{1}(*X, Y*)) *≤* *X*^{3}, LM(*q*_{2}_{,}_{2}(*X, Y*)) *≤* *X*.

Thus, if*a*_{1} = 0, then

*A*_{2} = *−a*_{2}+*b*_{1}*,*
*B*_{2} = *a*^{2}_{1}*−a*_{3}*−a*_{1}*s*_{2}*,*

*C*_{2} = *−a*^{2}_{1}*a*_{2}+*a*_{2}*a*_{3}*−a*^{2}_{1}*b*_{1}*−a*_{3}*b*_{1}+*a*_{1}*b*_{3}*−a*_{1}*s*_{1}+*a*_{1}*a*_{2}*s*_{2}+*a*^{2}_{1}*t*_{3}

(3*.*4)

by (3.1) and (3.2). And, if *a*_{1} = 0, then *A*_{2} = *−a*_{2}+*b*_{1}, *B*_{2} =*−a*_{3}, *C*_{2} = *a*_{2}*a*_{3}*−a*_{3}*b*_{1} by
(3.1), (3.2) and (3.3). These values are the same as those in (3.4) with *a*_{1} = 0.

For *h*_{3}(*X, Y*)*∈G*^{},

*h*_{3}(*X, Y*)*g*_{2}(*X, Y*) = *q*_{3}_{,}_{1}(*X, Y*)*g*_{1}(*X, Y*) +*q*_{3}_{,}_{2}(*X, Y*)*F*(*X, Y*) (3*.*5)
for*q*_{3}_{,}_{1}(*X, Y*)*, q*_{3}_{,}_{2}(*X, Y*)*∈K*[*X, Y*] with LM(*q*_{3}_{,}_{1}(*X, Y*))*≤X*^{3}, LM(*q*_{3}_{,}_{2}(*X, Y*))*≤X*, and
*h*_{3}(*X, Y*)*g*_{3}(*X, Y*) = *q*_{4}_{,}_{1}(*X, Y*)*g*_{1}(*X, Y*) +*q*_{4}_{,}_{2}(*X, Y*)*F*(*X, Y*) (3*.*6)
for *q*_{4}_{,}_{1}(*X, Y*)*, q*_{4}_{,}_{2}(*X, Y*) *∈* *K*[*X, Y*] with LM(*q*_{4}_{,}_{1}(*X, Y*)) *≤* *X*^{2}*Y*, LM(*q*_{4}_{,}_{2}(*X, Y*)) *≤* *Y*.
Thus, if*a*_{1} = 0, then

*A*_{3} = *a*_{1}^{2}*−b*_{2}*−a*_{1}*s*_{2}*,*

*B*_{3} = 2*a*_{1}*b*_{1}*−b*_{3}+*s*_{1}*−b*_{1}*s*_{2}*−a*_{1}*t*_{3}*,*

*C*_{3} = *−*2*a*_{1}*a*^{2}_{2} + 2*a*^{2}_{1}*a*_{3} + 2*a*_{1}*a*_{2}*b*_{1}*−a*_{1}*b*^{2}_{1}*−*3*a*^{2}_{1}*b*_{2}+*b*^{2}_{2}+*a*_{2}*b*_{3}*−b*_{1}*b*_{3}
+*s*_{0}+*a*^{2}_{2}*s*_{2}*−a*_{1}*a*_{3}*s*_{2}*−a*_{2}*b*_{1}*s*_{2} + 2*a*_{1}*b*_{2}*s*_{2}*−a*_{1}*t*_{2}+*a*_{1}*b*_{1}*t*_{3}

(3*.*7)

by (3.1) and (3.5). And, if *a*_{1} = 0, then *A*_{3} = *−b*_{2}, *B*_{3} = *−b*_{3} +*s*_{1} *−b*_{1}*s*_{2} and *C*_{3} =
*b*^{2}_{2}+*a*_{2}*b*_{3}*−b*_{1}*b*_{3}+*s*_{0}+*a*^{2}_{2}*s*_{2}*−a*_{2}*b*_{1}*s*_{2} by (3.1), (3.5) and (3.6). These values are the same
as those in (3.7) with *a*_{1} = 0.

Hence, we completely proved it. *2*

From now on, we use the following notation: For *i*= 1*,*2,
*I*_{i} : a normal ideal *ϕ*^{−1}(*L*(*∞ · ∞ −E*_{i}))*,*

*I*^{} : a normal ideal *ϕ*^{−1}(*L*(*∞ · ∞ −E*^{}))*,*
*I* : a normal ideal *ϕ*^{−1}(*L*(*∞ · ∞ −E*))*,*
*G*_{i} : a reduced Groebner basis for *I*_{i}*,*

*G*_{g} : a set *{f*_{i}(*X, Y*)*g*_{j}(*X, Y*)*, F*(*X, Y*)*|f*_{i}(*X, Y*)*∈G*_{1}*, g*_{j}(*X, Y*)*∈G*_{2}*},*
*G* : a reduced Groebner basis for *I,*

*H* : a reduced Groebner basis for *ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2}) =*ϕ*^{−1}(*L*(*∞ · ∞ −*(*E*_{1}+*E*_{2})))*,*
*h*_{1}(*X, Y*) : a polynomial with the smallest leading monomial in *H,*

*v*_{1}(*X, Y*) : a monic polynomial with the smallest leading monomial in *I*^{}*.*
The ﬁnal purpose of this section is to ﬁnd *G*for the given *G*_{1} and *G*_{2}.

We ﬁrst study a way of ﬁnding the reduced Groebner basis*H*for*ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2}) by using
the fact that *G*_{g} is a generating set of *ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2}). We have *δ*(*H*) = *n*_{1}+*n*_{2}. Thus, if
*δ*(*G*_{g}) *> n*_{1} +*n*_{2}, then *G*_{g} is not a Groebner basis and it is necessary to do division of
S-polynomials by the algorithm due to Buchberger. It is possible to omit the following
S-polynomials in *G*_{g}:

(a) *S*(*f*_{i}*g*_{j}*, f*_{i}*g*_{j}) for*f*_{i}*, f*_{i} *∈G*_{1}, *g*_{j}*, g*_{j} *∈G*_{2} with *i*=*i*^{} or *j* =*j*^{};
(b) *S*(*f, g*) for*f, g* *∈G*_{g} with lcm(LM(*f*)*,*LM(*g*)) = LM(*f*)LM(*g*);

(c) *S*(*f, g*) for *f, g∈G*_{g} with *h*=*f, g*in*G*_{g} such that*S*(*f, h*) and *S*(*g, h*) are divisible
by*G*_{g}, and LT(*h*) divides lcm(LT(*f*)*,*LT(*g*)).

Let *S* = *{S*_{1}*, . . . , S*_{m}*}* be the set of S-polynomials in *G*_{g} except those S-polynomials.

For *i* = 1*, . . . , m*, let *r*_{i} be the remainder of *S*_{i} on division by *G*_{g} *∪ {r*_{1}*, . . . , r*_{i}_{−1}*}*. Let
*G*_{g,}_{1} =*G*_{g} *∪ {r*_{1}*, . . . , r*_{m}*}*. Then

*n*_{1}+*n*_{2} *≤δ*(*G**g,*1)*< δ*(*G**g*)*.*

If*n*_{1}+*n*_{2} *< δ*(*G*_{g,}_{1}), it is needed to consider S-polynomials in*G*_{g,}_{1}. For every*r*_{i} = 0, it is
enough to consider the S-polynomials *S*(*r*_{i}*, f*) and*S*(*r*_{i}*, g*), where*f* (resp. *g*) is a nearest
element to *r*_{i} in the lower right-hand (resp. in the upper left-hand) as considering the
leading monomials by the above (c). By iterating this work until the value of *δ* decreases
to*n*_{1}+*n*_{2}, we get a Groebner basis. Thus the number of divisions to be done for getting
a Groebner basis for *ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2}) is *m*+ 2(*δ*(*G*_{g})*−*(*n*_{1}+*n*_{2})*−*1) at most.

Since we have *δ*(*H*) = *n*_{1} +*n*_{2} *≤* 6, *H* contains an element whose leading monomial
is smaller than *Y*^{3}. Thus, *D*^{} =*−*(*D*_{1}+*D*_{2}) + (*h*_{1}) for the polynomial *h*_{1}(*X, Y*) with the
smallest leading monomial in*H* by Proposition 3.3.1. Furthermore, we have

*I*^{} =*{v*(*X, Y*)*|v*(*X, Y*)*h*_{i}(*X, Y*)*∈ h*_{1}(*X, Y*)*, F*(*X, Y*) for all *h*_{i}(*X, Y*)*∈H}*
as shown in Section 3.3. Since *D*^{} is a normal divisor,*D*=*−D*^{}+ (*v*_{1}) and *v*_{1}(*X, Y*)*∈G*.

It follows that

*D*=*D*_{1}+*D*_{2}*−*(*h*_{1}) + (*v*_{1})*.*

Thus

*E* =*E*_{1}+*E*_{2} *−*(*h*_{1})^{+}+ (*v*_{1})^{+}

because the divisors *D*, *D*_{1}+*D*_{2}, (*h*_{1}) and (*v*_{1}) have no pole point but at inﬁnity. Since
*D*^{} = *−*(*D*_{1} +*D*_{2}) + (*h*_{1}), we have *n*^{} = deg(*h*_{1})^{+}*−*(*n*_{1} +*n*_{2}). Thus LM(*v*_{1}(*X, Y*)) is
determined if deg(*h*_{1})^{+} = *n*_{1} +*n*_{2} + 2. If deg(*h*_{1})^{+} = *n*_{1} +*n*_{2} + 2, then LM(*v*_{1}(*X, Y*))
is either *X* or *Y*. Further, LM(*v*_{1}(*X, Y*)) is determined by LM(*v*_{1}(*X, Y*)*h*_{i}(*X, Y*)) *∈*
LM(*h*_{1}(*X, Y*)*, F*(*X, Y*)) for all*h*_{i}(*X, Y*)*∈H*. Let*H* =*{h*_{1}(*X, Y*)*, . . . , h*_{t}(*X, Y*)*}*. Then
LM(*G*) is determined by LM(*H*) and *G* is obtained by a generating set for *I*, which is
given as follows.

Let *q*_{i,}_{1}(*X, Y*)*, q*_{i,}_{2}(*X, Y*)*∈K*[*X, Y*] satisfy

*v*_{1}(*X, Y*)*h*_{i}(*X, Y*) =*q*_{i,}_{1}(*X, Y*)*h*_{1}(*X, Y*) +*q*_{i,}_{2}(*X, Y*)*F*(*X, Y*)*.*

Then

(*v*_{1}) + (*h*_{i}) = (*q*_{i,}_{1}) + (*h*_{1})*.*

It follows that

(*v*_{1})^{+}+ (*h*_{i})^{+}= (*q*_{i,}_{1})^{+}+ (*h*_{1})^{+}*.*
Thus

(*q*_{i,}_{1})^{+}= (*v*_{1})^{+}+ (*h*_{i})^{+}*−*(*h*_{1})^{+}*.*

Since (*h*_{i})^{+}*≥E*_{1}+*E*_{2}, we have *q*_{i,}_{1} *∈L*(*∞ · ∞ −E*). Thus *q*_{i,}_{1}(*X, Y*)*∈I*.

If *f*(*X, Y*)*∈I*, then (*f*)^{+} *≥E*. It follows that
(*f*)^{+} *≥* *E*_{1}+*E*_{2}*−*(*h*_{1})^{+}+ (*v*_{1})^{+}

= min*{*(*h**i*)^{+}*−*(*h*_{1})^{+}+ (*v*_{1})^{+} *|i*= 1*, . . . , t}*

= min*{*(*q*_{i,}_{1})^{+} *|i*= 1*, . . . , t}.*

It implies that *f* *∈ q*_{1}_{,}_{1}*, . . . , q*_{t,}_{1}. Thus

*f*(*X, Y*) *∈* *ϕ*^{−1}(*q*_{1}_{,}_{1}*, . . . , q*_{t,}_{1})

= *q*_{1}_{,}_{1}(*X, Y*)*, . . . , q*_{t,}_{1}(*X, Y*)*, F*(*X, Y*)*.*
Hence *I* =*q*_{1}_{,}_{1}(*X, Y*)*, . . . , q*_{t,}_{1}(*X, Y*)*, F*(*X, Y*).

Now, we study the sum *D*_{1} +*D*_{2}. We assume that coeﬃcients *A*_{i}*, B*_{i}*, C*_{i}*, a*_{i}*, b*_{i}*, c*_{i} of
polynomials are elements of*K* and we assume that*S*_{i}*, r*_{i} are polynomials in*K*[*X, Y*]. For
a polynomial*f*(*X, Y*), we write *f* instead of *f*(*X, Y*).

**I***.***n**_{1} =**1***,***n**_{2} =**1**

Since deg (*D*_{1}+*D*_{2})^{+} = 2, *D*_{1} +*D*_{2} is a normal divisor by Theorem 4.1.2. Thus
*D*=*D*_{1}+*D*_{2}, and *G*is equal to *H*. If the reduced Groebner bases are

*G*_{1} = *{f*_{1}(*X, Y*) = *X*+*C*_{1}*, f*_{2}(*X, Y*) = *Y* +*C*_{2}*},*
*G*_{2} = *{g*_{1}(*X, Y*) =*X*+*c*_{1}*, g*_{2}(*X, Y*) =*Y* +*c*_{2}*},*

we have the following diagram on LM(*G*_{g}). The point of vacant circle denotes an element
in ∆(*G**g*), and ‘MD = (*i, j*) : *f*(*X, Y*)’ means LM(*f*(*X, Y*)) = *X*^{i}*Y*^{j} for a polynomial
*f*(*X, Y*)*∈G*_{g}.

e e u u

e u u u

u u u u

u u u u

0 1 2 3 *i*

1
2
3
*j*

(*i, j*)*↔X*^{i}*Y*^{j}

LM(*G**g*) = *{X*^{2}*, XY, Y*^{2}*, Y*^{3}*}*

MD =

(2*,*0) :*f*_{1}*g*_{1}
(1*,*1) :*f*_{1}*g*_{2}*, f*_{2}*g*_{1}
(0*,*2) :*f*_{2}*g*_{2}
(0*,*3) :*F*

It follows that*S* =*{S*_{1} =*S*(*f*_{1}*g*_{2}*, f*_{2}*g*_{1})*, S*_{2} =*S*(*F, f*_{2}*g*_{2})*}*with deg(*G*_{g}) =*n*_{1}+*n*_{2}+ 1. For
a nonzero *r*_{i} in*{r*_{1}*, r*_{2}*}*, LM(*r*_{i}) is either *X* or*Y*. The remainder of *S*_{1} on division by*G*_{g}
is

*r*_{1} = (*C*_{1}*−c*_{1})*Y* + (*−C*_{2}+*c*_{2})*X*+*C*_{1}*c*_{2}*−C*_{2}*c*_{1}*.*

Further, if *r*_{1} = 0, i.e. *G*_{1} =*G*_{2}, the remainder of*S*_{2} on division by *G*_{g} is
*r*_{2} =*F*_{Y}(*−C*_{1}*,−C*_{2})(*Y* +*C*_{2}) +*F*_{X}(*−C*_{1}*,−C*_{2})(*X*+*C*_{1})*,*

where *F*_{X} (resp. *F*_{Y}) denotes the partial derivative of *F*(*X, Y*) with respect to *X* (resp.

*Y*).

Since *δ*(*H*) = 2, we have the following diagrams on LM(*H*) = LM(*G*).

b r r r b r r r r r r r r r r r

*i*
*j*

(1)

LM(*H*) = LM(*G*)

=*{X, Y*^{2}*}*

b b r r r r r r r r r r r r r r

*i*
*j*

(2)

LM(*H*) = LM(*G*)

=*{Y, X*^{2}*}*
As a result, we have the following:

(i) If *G*_{1} =*G*_{2} with *C*_{1} =*c*_{1}, then*H* =*G*=*{*(*C*_{1}*−c*_{1})^{−1}*r*_{1}*, f*_{1}*g*_{1}*}*.
(ii) If *G*_{1} =*G*_{2} with *C*_{1} =*c*_{1}, then*H* =*G*=*{f*_{1}*, f*_{2}*g*_{2}*}*.

(iii) If *G*_{1} =*G*_{2} and *F*_{Y}(*−C*_{1}*,−C*_{2})= 0, then *H* =*G*=*{F*_{Y}(*−C*_{1}*,−C*_{2})^{−1}*r*_{2}*, f*_{1}*g*_{1}*}*.
(iv) If *G*_{1} =*G*_{2} and *F*_{Y}(*−C*_{1}*,−C*_{2}) = 0, then *H* =*G*=*{f*_{1}*, f*_{2}*g*_{2}*}*.

**II***.***n**_{1} =**1***,***n**_{2} =**2**

For a normal divisor *D*_{2} of pole degree 2, LM(*G*_{2}) is either *{X, Y*^{2}*}* or *{Y, X*^{2}*}*.

**1***.* **LM**(**G**_{2}) =*{***X***,***Y**^{2}*}*

If the reduced Groebner bases are

*G*_{1} = *{f*_{1}(*X, Y*) =*X*+*C*_{1}*, f*_{2}(*X, Y*) =*Y* +*C*_{2}*},*
*G*_{2} = *{g*_{1}(*X, Y*) =*X*+*c*_{1}*, g*_{2}(*X, Y*) =*Y*^{2}+*a*_{2}*Y* +*c*_{2}*},*

we have the following diagram on LM(*G*_{g}).

e e u u

e u u u

e u u u

u u u u

0 1 2 3 *i*

1
2
3
*j*

LM(*G*_{g}) = *{X*^{2}*, XY, XY*^{2}*, Y*^{3}*}*

(*i, j*)*↔X*^{i}*Y*^{j}

MD = (0*,*3) :*f*_{2}*g*_{2}*, F*
(1*,*2) :*f*_{1}*g*_{2}
(1*,*1) :*f*_{2}*g*_{1}
(2*,*0) :*f*_{1}*g*_{1}

It follows that*S* =*{S*_{1} =*S*(*f*_{1}*g*_{2}*, f*_{2}*g*_{1})*, S*_{2} =*S*(*F, f*_{2}*g*_{2})*}*with *δ*(*G*_{g}) =*n*_{1}+*n*_{2}+ 1. Thus,
for *r*_{i} = 0 in *{r*_{1}*, r*_{2}*}*, LM(*r*_{i}) is either *X* or *Y*^{2}. The coeﬃcient of *Y*^{2} in *r*_{1} is *C*_{1} *−c*_{1}.
Further, if *r*_{1} = 0, the coeﬃcient of *Y*^{2} in *r*_{2} is *−C*_{2}*−a*_{2}. It follows that *H* contains an
element whose leading monomial is *X* if and only if *C*_{1} =*c*_{1} and *C*_{2} =*−a*_{2}.

Since*δ*(*H*) = 3 with ∆(*H*)*⊂ {*1*, X, Y, Y*^{2}*}*, we have the following diagrams on LM(*H*),
which are followed by LM(*G*).

b r r r b r r r b r r r r r r r

*i*
*j*

(1)

LM(*H*)

=*{X, Y*^{3}*}*

*⇔*LM(*G*)

=*{*1*}*

b b r r b r r r r r r r r r r r

*i*
*j*

(2)

LM(*H*) = LM(*G*)

=*{X*^{2}*, XY, Y*^{2}*}*

As a result, *H* and *G*are as follows:

(i) If *C*_{1} =*c*_{1} and *C*_{2} =*−a*_{2}, then *H*=*{f*_{1}*, f*_{2}*g*_{2}*}* and *G*=*{*1*}*.

(ii) If *C*_{1} =*c*_{1} and *C*_{2} =*−a*_{2}, then*H* =*G*=*{f*_{1}*g*_{1}*, f*_{2}*g*_{1}*,−*(*C*_{2}+*a*_{2})^{−1}*r*_{2}*}*.
(iii) If *C*_{1} =*c*_{1}, then *H* =*G*=*{f*_{1}*g*_{1}*, f*_{2}*g*_{1}*,*(*C*_{1}*−c*_{1})^{−1}*r*_{1}*}*.

**2***.* **LM**(**G**_{2}) =*{***Y***,***X**^{2}*}*

If the reduced Groebner bases are

*G*_{1} = *{f*_{1}(*X, Y*) =*X*+*C*_{1}*, f*_{2}(*X, Y*) =*Y* +*C*_{2}*},*

*G*_{2} = *{g*_{1}(*X, Y*) =*Y* +*b*_{1}*X*+*c*_{1}*, g*_{2}(*X, Y*) =*X*^{2}+*b*_{2}*X*+*c*_{2}*},*

we have the following diagram on LM(*G*_{g}).

e e e u

e u u u

u u u u

u u u u

0 1 2 3 *i*

1
2
3
*j*

LM(*G*_{g}) = *{XY, Y*^{2}*, X*^{3}*, X*^{2}*Y, Y*^{3}*}*
(*i, j*)*↔X*^{i}*Y*^{j}

MD = (0*,*3) :*F*
(0*,*2) :*f*_{2}*g*_{1}

(1*,*1) :*f*_{1}*g*_{1} (2*,*1) :*f*_{2}*g*_{2}
(3*,*0) :*f*_{1}*g*_{2}

It follows that *S* =*{S*_{1} =*S*(*f*_{1}*g*_{1}*, f*_{2}*g*_{2})*, S*_{2} = *S*(*F, f*_{2}*g*_{1})*}* with *δ*(*G*_{g}) = *n*_{1}+*n*_{2}+ 1. For
*r*_{i} = 0 in*{r*_{1}*, r*_{2}*}*, LM(*r*_{i}) is either*Y* or*X*^{2}. The coeﬃcient of*X*^{2} in*r*_{1} is*−g*_{1}(*−C*_{1}*,−C*_{2}).

Further, if *r*_{1} = 0, the coeﬃcient of *X*^{2} in*r*_{2} is the remainder on division of the quotient
*F*(*X,−b*_{1}*X−c*_{1})*/g*_{2} by*f*_{1}.

Since*δ*(*H*) = 3 with ∆(*H*)*⊂ {*1*, X, Y, X*^{2}*}*, we have the following diagrams on LM(*H*),
which are followed by LM(*G*).

b b b r r r r r r r r r r r r r

*i*
*j*

(1)

LM(*H*)

=*{Y, X*^{3}*}*

*⇔*LM(*G*)

=*{X, Y*^{2}*}*

b b r r b r r r r r r r r r r r

*i*
*j*

(2)

LM(*H*) = LM(*G*)

=*{X*^{2}*, XY, Y*^{2}*}*

As a result, we have *H* and *G* as follows:

(i) If *g*_{1}(*−C*_{1}*,−C*_{2}) = 0 and *F*(*X,−b*_{1}*X−c*_{1}) is divisible by *f*_{1}*g*_{2}, then *H* = *{h*_{1} =
*g*_{1}*, h*_{2} =*f*_{1}*g*_{2}*}*and LM(*G*) =*{X, Y*^{2}*}*. For the polynomial *v*_{1}, we have

*v*_{1}*h*_{2} =*q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}(*F* *−Y*^{2}*h*_{1})

for *q*_{2}_{,}_{1}*, q*_{2}_{,}_{2} *∈* *K*[*X, Y*] with LT(*v*_{1}) = *X,*LM(*q*_{2}_{,}_{1}) *≤* *XY, q*_{2}_{,}_{2} = 1 since *{h*_{1}*, F* *−Y*^{2}*h*_{1}*}*
is a Groebner basis for *h*_{1}*, F* and *v*_{1}*h*_{2} *∈ h*_{1}*, F*. It follows that *{v*_{1}*, Y*^{2} *−q*_{2}_{,}_{1}*}* is a
Groebner basis, whose elements are monic polynomials, for *I*. Thus *G*=*{v*_{1}*, v*_{2}*}* for the
remainder *v*_{2} of *Y*^{2} *−q*_{2}_{,}_{1} on division by *v*_{1}.

(ii) If *g*_{1}(*−C*_{1}*,−C*_{2}) = 0 and *F*(*X,−b*_{1}*X−c*_{1}) is not divisible by *f*_{1}*g*_{2}, then*H* =*G*=
*{r*_{2}_{,m}*, f*_{1}*g*_{1}*−b*_{1}*r*_{2}_{,m}*, f*_{2}*g*_{1}*−b*_{1}*f*_{1}*g*_{1}+*b*^{2}_{1}*r*_{2}_{,m}*}*.

(iii) If *g*_{1}(*−C*_{1}*,−C*_{2})= 0, then *H* =*G*=*{r*_{1}_{,m}*, f*_{1}*g*_{1}*−b*_{1}*r*_{1}_{,m}*, f*_{2}*g*_{1}*−b*_{1}*f*_{1}*g*_{1}+*b*^{2}_{1}*r*_{1}_{,m}*}*.

**III***.***n**_{1} =**1***,***n**_{2} =**3**

If the reduced Groebner bases are

*G*_{1} = *{f*_{1}(*X, Y*) = *X*+*C*_{1}*, f*_{2}(*X, Y*) = *Y* +*C*_{2}*},*

*G*_{2} = *{g*_{1}(*X, Y*) = *X*^{2}+*a*_{1}*Y* +*b*_{1}*X*+*c*_{1}*, g*_{2}(*X, Y*) = *XY* +*a*_{2}*Y* +*b*_{2}*X*+*c*_{2}*,*
*g*_{3}(*X, Y*) =*Y*^{2}+*a*_{3}*Y* +*b*_{3}*X*+*c*_{3}*},*

we have the following diagram on LM(*G*_{g}).

e e e u

e e u u

e u u u

u u u u

0 1 2 3 *i*

1
2
3
*j*

LM(*G*_{g}) = *{X*^{3}*, X*^{2}*Y, XY*^{2}*, Y*^{3}*}*

(*i, j*)*↔X*^{i}*Y*^{j}

MD = (0*,*3) :*f*_{2}*g*_{3}*, F*
(1*,*2) :*f*_{1}*g*_{3}*, f*_{2}*g*_{2}
(2*,*1) :*f*_{1}*g*_{2}*, f*_{2}*g*_{1}
(3*,*0) :*f*_{1}*g*_{1}

It follows that*S* =*{S*_{1} =*S*(*f*_{1}*g*_{2}*, f*_{2}*g*_{1})*, S*_{2} =*S*(*f*_{1}*g*_{3}*, f*_{2}*g*_{2})*, S*_{3} =*S*(*F, f*_{2}*g*_{3})*}*with *δ*(*G*_{g}) =
*n*_{1} +*n*_{2} + 2. For *G*_{g,}_{1} = *G*_{g} *∪ {r*_{1}*, r*_{2}*, r*_{3}*}*, *δ*(*G*_{g,}_{1}) is either 4 or 5. If *δ*(*G*_{g,}_{1}) = 4, then
*G*_{g,}_{1} is a Groebner basis. If *δ*(*G*_{g,}_{1}) = 5, then there is exactly one nonzero polynomial in
*{r*_{1}*, r*_{2}*, r*_{3}*}*. For*r*_{i} = 0, LM(*r*_{i}) is*X*^{2},*XY* or*Y*^{2} and it is enough to consider the following
S-polynomials in *G*_{g,}_{1}:

(i) *S*(*r**i**, f*_{1}*g*_{1}) and*S*(*r**i**, f*_{1}*g*_{2}) if LM(*r**i*) =*X*^{2};
(ii) *S*(*r*_{i}*, f*_{1}*g*_{2}) and *S*(*r*_{i}*, f*_{1}*g*_{3}) if LM(*r*_{i}) =*XY*;
(iii) *S*(*r*_{i}*, f*_{1}*g*_{3}) and *S*(*r*_{i}*, f*_{2}*g*_{3}) if LM(*r*_{i}) =*Y*^{2}.

For a nonzero remainder *r* of these S-polynomials on division by *G*_{g,}_{1}, *G*_{g,}_{1} *∪ {r}* is a
Groebner basis for*ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2}) since*δ*(*G*_{g,}_{1}) =*n*_{1}+*n*_{2}+ 1. Thus, the number of divisions
to be done for getting a Groebner basis for *ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2}) is 5 at most.

Since *δ*(*H*) = 4 with ∆(*H*) *⊂ {*1*, X, Y, X*^{2}*, XY, Y*^{2}*}*, we have the following diagrams
on LM(*H*), which are followed by LM(*G*).

b b r r b r r r b r r r r r r r

*i*
*j*

(1)

LM(*H*)

=*{X*^{2}*, XY, Y*^{3}*}*

*⇔*LM(*G*)

=*{X, Y}*

b b r r b b r r r r r r r r r r

*i*
*j*

(2)

LM(*H*)

=*{X*^{2}*, Y*^{2}*}*

*⇔*LM(*G*)

=*{Y, X*^{2}*}*

b b b r b r r r r r r r r r r r

*i*
*j*

(3)

LM(*H*)

=*{XY, Y*^{2}*, X*^{3}*}*

*⇔*LM(*G*)

=*{X*^{2}*, XY, Y*^{2}*}*
In the case of (1), *n*^{} = deg (*h*_{1})^{+}*−*(*n*_{1} +*n*_{2}) = 2. It follows that LM(*v*_{1}) is either
*X* or *Y*. Let *H* = *{h*_{1}*, h*_{2}*, h*_{3}*}* with LM(*h*_{1}) = *X*^{2}*,*LM(*h*_{2}) = *XY,*LM(*h*_{3}) = *Y*^{3}. Then
*{h*_{1}*, F}*is a Groebner basis for*h*_{1}*, F*since lcm(LM(*h*_{1})*,*LM(*F*)) = LM(*h*_{1})LM(*F*). Thus
LM(*v*_{1}*h*_{i}) *∈ X*^{2}*, Y*^{3} for all *h*_{i} *∈* *H*. It follows that LM(*v*_{1}) = *X*. Further, *n* = 1 and
LM(*G*) =*{X, Y}*. Since *v*_{1}*h*_{2} *∈ h*_{1}*, F*, we have

*v*_{1}*h*_{2} =*q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}*F*

for *q*_{2}_{,}_{1}*, q*_{2}_{,}_{2} *∈* *K*[*X, Y*] with LT(*q*_{2}_{,}_{1}) =*Y, q*_{2}_{,}_{2} = 0. It follows that *{v*_{1}*, q*_{2}_{,}_{1}*}* is a Groebner
basis for *I*. Thus *G*=*{v*_{1}*, v*_{2}*}* for the remainder *v*_{2} of *q*_{2}_{,}_{1} on division by*v*_{1}.

In the case of (2), *n*^{} = deg (*h*_{1})^{+}*−*(*n*_{1}+*n*_{2}) = 2. Let *H* =*{h*_{1}*, h*_{2}*}* with LM(*h*_{1}) =
*X*^{2}*,*LM(*h*_{2}) = *Y*^{2}. Then *{h*_{1}*, F}* is a Groebner basis for *h*_{1}*, F*. Thus LM(*v*_{1}*h*_{i}) *∈*
*X*^{2}*, Y*^{3} for all *h*_{i} *∈* *H*. It follows that LM(*v*_{1}) = *Y*. Further, *n* = 2 and LM(*G*) =
*{Y, X*^{2}*}*. Since *v*_{1}*h*_{2} *∈ h*_{1}*, F*, we have

*v*_{1}*h*_{2} =*q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}*F*

for *q*_{2}_{,}_{1}*, q*_{2}_{,}_{2} *∈* *K*[*X, Y*] with LT(*q*_{2}_{,}_{1}) = *−X*^{2}*, q*_{2}_{,}_{2} = 1. It follows that *{v*_{1}*,−q*_{2}_{,}_{1}*}* is a
Groebner basis for *I*. Thus *G*=*{v*_{1}*, v*_{2}*}* for the remainder *v*_{2} of *−q*_{2}_{,}_{1} on division by *v*_{1}.

In the case of (3), *n*^{} = deg (*h*_{1})^{+}*−*(*n*_{1}+*n*_{2}) = 3. Thus LM(*v*_{1}) =*X*^{2} and LM(*G*) =
*{X*^{2}*, XY, Y*^{2}*}*. Let *H* = *{h*_{1}*, h*_{2}*, h*_{3}*}* with LM(*h*_{1}) = *XY,*LM(*h*_{2}) = *Y*^{2}*,*LM(*h*_{3}) = *X*^{3}.
Then *{h*_{1}*, F, XF* *−Y*^{2}*h*_{1}*}* is a Groebner basis for *h*_{1}*, F*. Since *v*_{1}*h*_{i} *∈ h*_{1}*, F* for all
*h*_{i} *∈H*, we have

*v*_{1}*h*_{2} =*q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}*F* +*q*_{2}_{,}_{3}(*XF* *−Y*^{2}*h*_{1})

for *q*_{2}_{,}_{1}*, q*_{2}_{,}_{2}*, q*_{2}_{,}_{3} *∈K*[*X, Y*] with LT(*q*_{2}_{,}_{1}) = *XY,*LM(*q*_{2}_{,}_{2})*≤*1*, q*_{2}_{,}_{3} = 0, and
*v*_{1}*h*_{3} =*q*_{3}*,*1*h*_{1}+*q*_{3}*,*2*F* +*q*_{3}*,*3(*XF* *−Y*^{2}*h*_{1})

for *q*_{3}_{,}_{1}*, q*_{3}_{,}_{2}*, q*_{3}_{,}_{3} *∈* *K*[*X, Y*] with LM(*q*_{3}_{,}_{1}) *≤* *XY,*LM(*q*_{3}_{,}_{2}) *≤* 1*, q*_{3}_{,}_{3} = 1. It follows that
*{v*_{1}*, q*_{2}*,*1*, Y*^{2}*−q*_{3}*,*1*}* is a Groebner basis for*I*. Thus *G* =*{v*_{1}*, v*_{2}*, v*_{3}*}* for the remainder *v*_{2}
of *q*_{2}_{,}_{1} on division by*v*_{1} and the remainder*v*_{3} of *Y*^{2}*−q*_{3}_{,}_{1} on division by *{v*_{1}*, v*_{2}*}*.

**Remark.** We have another way to ﬁnd *H* according to the relation between *G*_{1} and *G*_{2}.
We give it in Appendix.

**IV***.* **n**_{1} =**2***,***n**_{2} =**2**

**1***.* **LM**(**G**_{1}) =*{***X***,***Y**^{2}*},***LM**(**G**_{2}) = *{***X***,***Y**^{2}*}*

If the reduced Groebner bases are

*G*_{1} = *{f*_{1}(*X, Y*) =*X*+*C*_{1}*, f*_{2}(*X, Y*) =*Y*^{2} +*A*_{2}*Y* +*C*_{2}*},*
*G*_{2} = *{g*_{1}(*X, Y*) =*X*+*c*_{1}*, g*_{2}(*X, Y*) = *Y*^{2}+*a*_{2}*Y* +*c*_{2}*},*
we have the following diagram on LM(*G**g*).

e e u u

e e u u

e u u u

u u u u

u u u u

0 1 2 3 *i*

1
2
3
4
*j*

LM(*G*_{g}) = *{X*^{2}*, XY*^{2}*, Y*^{3}*, Y*^{4}*}*

(*i, j*)*↔X*^{i}*Y*^{j}

MD = (0*,*4) :*f*_{2}*g*_{2}
(0*,*3) :*F*

(1*,*2) :*f*_{1}*g*_{2}*, f*_{2}*g*_{1}

(2*,*0) :*f*_{1}*g*_{1}

It follows that *S* = *{S*_{1} = *S*(*f*_{1}*g*_{2}*, f*_{2}*g*_{1})*, S*_{2} = *S*(*F, f*_{1}*g*_{2})*, S*_{3} =*S*(*F, f*_{2}*g*_{2})*}* with *δ*(*G*_{g}) =
*n*_{1}+*n*_{2}+ 1. For a nonzero*r*_{i} in*{r*_{1}*, r*_{2}*, r*_{3}*}*,*G*_{g}*∪ {r*_{i}*}*is a Groebner basis for *ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2})
with LM(*r*_{i}) =*XY* or *Y*^{2} since *δ*(*G*_{g}) =*n*_{1}+*n*_{2}+ 1.

Since *δ*(*H*) = 4 with ∆(*H*) *⊂ {*1*, X, Y, XY, Y*^{2}*}*, we have the following diagrams on
LM(*H*) with the same result on *G* corresponding to *H* as that of**III***.* **n**_{1} =**1***,***n**_{2} =**3**.

b b r r b r r r b r r r r r r r

*i*
*j*

(1)

LM(*H*)

=*{X*^{2}*, XY, Y*^{3}*}*

*⇔*LM(*G*)

=*{X, Y}*

b b r r b b r r r r r r r r r r

*i*
*j*

(2)

LM(*H*)

=*{X*^{2}*, Y*^{2}*}*

*⇔*LM(*G*)

=*{Y, X*^{2}*}*

**2***.* **LM**(**G**_{1}) =*{***X***,***Y**^{2}*},***LM**(**G**_{2}) = *{***Y***,***X**^{2}*}*

If the reduced Groebner bases are

*G*_{1} = *{f*_{1}(*X, Y*) =*X*+*C*_{1}*, f*_{2}(*X, Y*) =*Y*^{2}+*A*_{2}*Y* +*C*_{2}*},*
*G*_{2} = *{g*_{1}(*X, Y*) =*Y* +*b*_{1}*X*+*c*_{1}*, g*_{2}(*X, Y*) =*X*^{2}+*b*_{2}*X*+*c*_{2}*},*
we have the following diagram on LM(*G*_{g}).

e e e u

e u u u

e u u u

u u u u

0 1 2 3 *i*

1
2
3
*j*

LM(*G*_{g}) = *{XY, X*^{3}*, Y*^{3}*, X*^{2}*Y*^{2}*}*

(*i, j*)*↔X*^{i}*Y*^{j}

MD = (0*,*3) :*f*_{2}*g*_{1}*, F*
(2*,*2) :*f*_{2}*g*_{2}
(1*,*1) :*f*_{1}*g*_{1}
(3*,*0) :*f*_{1}*g*_{2}

It follows that *S* =*{S*_{1} =*S*(*F, f*_{2}*g*_{1})*, S*_{2} =*S*(*f*_{2}*g*_{2}*, f*_{1}*g*_{1})*}*with *δ*(*G*_{g}) = *n*_{1}+*n*_{2}+ 1. For a
nonzero*r**i* in *{r*_{1}*, r*_{2}*}*,*G**g**∪ {r**i**}* is a Groebner basis for *ϕ*^{−1}(*I**D*_{1}+*D*_{2}).

Since *δ*(*H*) = 4 with ∆(*H*) *⊂ {*1*, X, Y, X*^{2}*, Y*^{2}*}*, we have the following diagrams on

LM(*H*) with the same result on *G* corresponding to *H* as that of**III***.* **n**_{1} =**1***,***n**_{2} =**3**.

b b r r b r r r b r r r r r r r

*i*
*j*

(1)

LM(*H*)

=*{X*^{2}*, XY, Y*^{3}*}*

*⇔*LM(*G*)

=*{X, Y}*

b b b r b r r r r r r r r r r r

*i*
*j*

(2)

LM(*H*)

=*{XY, Y*^{2}*, X*^{3}*}*

*⇔*LM(*G*)

=*{X*^{2}*, XY, Y*^{2}*}*

**3***.* **LM**(**G**_{1}) =*{***Y***,***X**^{2}*},***LM**(**G**_{2}) = *{***Y***,***X**^{2}*}*
If the reduced Groebner bases are

*G*_{1} = *{f*_{1}(*X, Y*) =*Y* +*B*_{1}*X*+*C*_{1}*, f*_{2}(*X, Y*) =*X*^{2}+*B*_{2}*X*+*C*_{2}*},*
*G*_{2} = *{g*_{1}(*X, Y*) = *Y* +*b*_{1}*X*+*c*_{1}*, g*_{2}(*X, Y*) =*X*^{2}+*b*_{2}*X*+*c*_{2}*},*
we have the following diagram on LM(*G**g*).

e e e e u

e e u u u

u u u u u

u u u u u

0 1 2 3 4 *i*

1
2
3
*j*

LM(*G*_{g}) = *{Y*^{2}*, X*^{2}*Y, X*^{4}*, Y*^{3}*}*

(*i, j*)*↔X*^{i}*Y*^{j}

MD = (0*,*3) :*F*
(0*,*2) :*f*_{1}*g*_{1}
(2*,*1) :*f*_{1}*g*_{2}*, f*_{2}*g*_{1}
(4*,*0) :*f*_{2}*g*_{2}

It follows that*S* =*{S*_{1} =*S*(*F, f*_{1}*g*_{3})*, S*_{2} =*S*(*f*_{2}*g*_{2}*, f*_{1}*g*_{1})*, S*_{3} =*S*(*f*_{2}*g*_{3}*, f*_{1}*g*_{2})*}*with *δ*(*G*_{g}) =
*n*_{1}+*n*_{2}+ 2. For*G**g,*1 =*G**g**∪ {r*_{1}*, r*_{2}*, r*_{3}*}*,*δ*(*G**g,*1) is either 5 or 6. If *δ*(*G**g,*1) = 5, then*G**g,*1

is a Groebner basis for *ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2}). If *δ*(*G*_{g,}_{1}) = 6, then there is exactly one nonzero
polynomial in *{r*_{1}*, r*_{2}*, r*_{3}*}*. For *r*_{i} = 0, LM(*r*_{i}) =*XY* or *X*^{3}, and it is enough to consider
the following S-polynomials:

(i) *S*(*f*_{1}*g*_{1}*, r*_{i}) and*S*(*f*_{1}*g*_{2}*, r*_{i}) if LM(*r*_{i}) =*XY*;
(ii) *S*(*f*_{2}*g*_{1}*, r*_{i}) and *S*(*f*_{2}*g*_{2}*, r*_{i}) if LM(*r*_{i}) =*X*^{3}.

Then, for a nonzero remainder *r* of these S-polynomials on division by *G*_{g,}_{1}, *G*_{g,}_{1}*∪ {r}*

is a Groebner basis for *ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2}) since *δ*(*G*_{g,}_{1}) = *n*_{1} +*n*_{2} + 1. Thus, the number of
divisions to be done for getting a Groebner basis for *ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2}) is 5 at most.

Since *δ*(*H*) = 4 with ∆(*H*) *⊂ {*1*, X, Y, X*^{2}*, XY, X*^{3}*}*, we have the following diagrams
on LM(*H*) with the same result on *G* corresponding to *H* as that of **III***.* **n**_{1} =**1***,***n**_{2} =**3**.

b b b b r r r r r r r r r r r r r r r r

*i*
*j*

(1)

LM(*H*)

=*{Y, X*^{4}*}*

*⇔*LM(*G*)

=*{*1*}*

b b r r r b b r r r r r r r r r r r r r

*i*
*j*

(2)

LM(*H*)

=*{X*^{2}*, Y*^{2}*}*

*⇔*LM(*G*)

=*{Y, X*^{2}*}*

b b b r r b r r r r r r r r r r r r r r

*i*
*j*

(3)

LM(*H*)

=*{XY, Y*^{2}*, X*^{3}*}*

*⇔*LM(*G*)

=*{X*^{2}*, XY, Y*^{2}*}*

**V***.* **n**_{1} =**2***,***n**_{2} =**3**

**1***.* **LM**(**G**_{1}) =*{***X***,***Y**^{2}*}*

If the reduced Groebner bases are

*G*_{1} = *{f*_{1}(*X, Y*) = *X*+*C*_{1}*, f*_{2}(*X, Y*) = *Y*^{2}+*A*_{2}*Y* +*C*_{2}*},*

*G*_{2} = *{g*_{1}(*X, Y*) = *X*^{2}+*a*_{1}*Y* +*b*_{1}*X*+*c*_{1}*, g*_{2}(*X, Y*) = *XY* +*a*_{2}*Y* +*b*_{2}*X*+*c*_{2}*,*
*g*_{3}(*X, Y*) =*Y*^{2}+*a*_{3}*Y* +*b*_{3}*X*+*c*_{3}*},*

we have the following diagram on LM(*G**g*).

e e e u

e e u u

e u u u

u u u u

u u u u

0 1 2 3 *i*

1
2
3
4
*j*

LM(*G*_{g}) = *{X*^{3}*, X*^{2}*Y, XY*^{2}*, Y*^{3}*, X*^{2}*Y*^{2}*, XY*^{3}*, Y*^{4}*}*
(*i, j*)*↔X*^{i}*Y*^{j}

MD = (0*,*4) :*f*_{2}*g*_{3}

(0*,*3) :*F* (1*,*3) : *f*_{2}*g*_{2}
(1*,*2) :*f*_{1}*g*_{3} (2*,*2) : *f*_{2}*g*_{1}
(2*,*1) :*f*_{1}*g*_{2}

(3*,*0) :*f*_{1}*g*_{1}

It follows that *S* = *{S*_{1} = *S*(*f*_{2}*g*_{1}*, f*_{1}*g*_{3})*, S*_{2} = *S*(*f*_{2}*g*_{2}*, f*_{1}*g*_{3})*, S*_{3} = *S*(*f*_{2}*g*_{2}*, F*)*, S*_{4} =
*S*(*f*_{2}*g*_{3}*, F*)*}* with *δ*(*G*_{g}) = *n*_{1} + *n*_{2} + 1. For *r*_{i} = 0 in *{r*_{1}*, r*_{2}*, r*_{3}*, r*_{4}*}*, *G*_{g} *∪ {r*_{i}*}* is a
Groebner basis for *ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2}).

Since *δ*(*H*) = 5 with ∆(*H*) *⊂ {*1*, X, Y, X*^{2}*, XY, Y*^{2}*}*, we have the following diagrams
on LM(*H*), which are followed by LM(*G*).

b b r r b b r r b r r r r r r r r r r r

*i*
*j*

(1)

LM(*H*)

=*{X*^{2}*, XY*^{2}*, Y*^{3}*}*

*⇔*LM(*G*)

=*{X, Y*^{2}*}*

b b b r b r r r b r r r r r r r r r r r

*i*
*j*

(2)

LM(*H*)

=*{XY, X*^{3}*, Y*^{3}*}*

*⇔*LM(*G*)

=*{Y, X*^{2}*}*

b b b r b b r r r r r r r r r r r r r r

*i*
*j*

(3)

LM(*H*)

=*{Y*^{2}*, X*^{3}*, X*^{2}*Y}*

*⇔*LM(*G*)

=*{X*^{2}*, XY, Y*^{2}*}*
In the case of (1), *n*^{} = 1 and LM(*v*_{1}) = *X*. Further, *n* = 2 and LM(*G*) =*{X, Y*^{2}*}*.
Let *H* =*{h*_{1}*, h*_{2}*, h*_{3}*}* with LM(*h*_{1}) =*X*^{2}*,*LM(*h*_{2}) =*XY*^{2}*,*LM(*h*_{3}) =*Y*^{3}. Then *{h*_{1}*, F}* is
a Groebner basis for *h*_{1}*, F*. Since *v*_{1}*h*_{2} *∈ h*_{1}*, F*, we have

*v*_{1}*h*_{2} =*q*_{2}*,*1*h*_{1}+*q*_{2}*,*2*F*

for *q*_{2}_{,}_{1}*, q*_{2}_{,}_{2} *∈* *K*[*X, Y*] with LT(*q*_{2}_{,}_{1}) = *Y*^{2}*,*LM(*q*_{2}_{,}_{2}) *≤* 1. It follows that *{v*_{1}*, q*_{2}_{,}_{1}*}* is a
Groebner basis for *I*. Thus *G*=*{v*_{1}*, v*_{2}*}* for the remainder *v*_{2} of *q*_{2}_{,}_{1} on division by *v*_{1}.

In the case of (2), *n*^{} = 2. It follows that LM(*v*_{1}) is either*X* or*Y*. Let*H*=*{h*_{1}*, h*_{2}*, h*_{3}*}*
with LM(*h*_{1}) =*XY,*LM(*h*_{2}) = *X*^{3}*,*LM(*h*_{3}) =*Y*^{3}. Then*{h*_{1}*, F, XF−Y*^{2}*h*_{1}*}*is a Groebner
basis for*h*_{1}*, F*. Since LM(*v*_{1}*h*_{i})*∈ XY, X*^{3}*, Y*^{3} for all*h*_{i} *∈H*, we have LM(*v*_{1}) =*Y*. It
follows that *n*= 2 and LM(*G*) =*{Y, X*^{2}*}*. Since*v*_{1}*h*_{2} *∈ h*_{1}*, F*, we have

*v*_{1}*h*_{2} =*q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}*F* +*q*_{2}_{,}_{3}(*XF* *−Y*^{2}*h*_{1})

for *q*_{2}_{,}_{1}*, q*_{2}_{,}_{2}*, q*_{2}_{,}_{3} *∈* *K*[*X, Y*] with LT(*q*_{2}_{,}_{1}) = *X*^{2}*,*LM(*q*_{2}_{,}_{2}) *≤* 1*, q*_{2}_{,}_{3} = 0. It follows that
*{v*_{1}*, q*_{2}_{,}_{1}*}* is a Groebner basis for *I*. Thus *G* = *{v*_{1}*, v*_{2}*}* for the remainder *v*_{2} of *q*_{2}_{,}_{1} on
division by*v*_{1}.

In the case of (3), *n*^{} = 3 and LM(*v*_{1}) =*X*^{2}. Further, LM(*G*) = *{X*^{2}*, XY, Y*^{2}*}*. Let
*H* =*{h*_{1}*, h*_{2}*, h*_{3}*}* with LM(*h*_{1}) =*Y*^{2}*,*LM(*h*_{2}) = *X*^{3}*,*LM(*h*_{3}) =*X*^{2}*Y*. Then *{h*_{1}*, F* *−Y h*_{1}*}*

is a Groebner basis for *h*_{1}*, F*. Since*v*_{1}*h*_{2} *∈ h*_{1}*, F*, we have
*v*_{1}*h*_{2} =*q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}(*F* *−Y h*_{1})
for *q*_{2}*,*1*, q*_{2}*,*2 *∈K*[*X, Y*] with LM(*q*_{2}*,*1)*≤X*^{2}*,*LT(*q*_{2}*,*2) = *X*, and

*v*_{1}*h*_{3} =*q*_{3}_{,}_{1}*h*_{1}+*q*_{3}_{,}_{2}(*F* *−Y h*_{1})

for *q*_{3}_{,}_{1}*, q*_{3}_{,}_{2} *∈* *K*[*X, Y*] with LM(*q*_{3}_{,}_{1}) *≤* *XY,*LT(*q*_{3}_{,}_{2}) = *Y*. It follows that *{v*_{1}*, q*_{2}_{,}_{2}*Y* *−*
*q*_{2}*,*1*, q*_{3}*,*2*Y* *−q*_{3}*,*1*}* is a Groebner basis for *I*. Thus*G*=*{v*_{1}*, v*_{2}*, v*_{3}*}*for the remainder *v*_{2} of
*q*_{2}_{,}_{2}*Y* *−q*_{2}_{,}_{1} on division by*v*_{1} and the remainder *v*_{3} of *q*_{3}_{,}_{2}*Y* *−q*_{3}_{,}_{1} on division by *{v*_{1}*, v*_{2}*}*.

**2***.* **LM**(**G**_{1}) =*{***Y***,***X**^{2}*}*

If the reduced Groebner bases are

*G*_{1} = *{f*_{1}(*X, Y*) = *Y* +*B*_{1}*X*+*C*_{1}*, f*_{2}(*X, Y*) =*X*^{2}+*B*_{2}*X*+*C*_{2}*},*

*G*_{2} = *{g*_{1}(*X, Y*) = *X*^{2}+*a*_{1}*Y* +*b*_{1}*X*+*c*_{1}*, g*_{2}(*X, Y*) = *XY* +*a*_{2}*Y* +*b*_{2}*X*+*c*_{2}*,*
*g*_{3}(*X, Y*) =*Y*^{2}+*a*_{3}*Y* +*b*_{3}*X*+*c*_{3}*},*

we have the following diagram on LM(*G*_{g}).

e e e e u

e e u u u

e u u u u

u u u u u

0 1 2 3 4 *i*

1
2
3
*j*

LM(*G*_{g}) = *{X*^{2}*Y, XY*^{2}*, X*^{4}*, Y*^{3}*, X*^{3}*Y, X*^{2}*Y*^{2}*}*
(*i, j*)*↔X*^{i}*Y*^{j}

MD = (0*,*3) : *f*_{1}*g*_{3}*, F*

(1*,*2) : *f*_{1}*g*_{2} (2*,*2) :*f*_{2}*g*_{3}
(2*,*1) : *f*_{1}*g*_{1} (3*,*1) :*f*_{2}*g*_{2}
(4*,*0) : *f*_{2}*g*_{1}

It follows that*S* =*{S*_{1} =*S*(*F, f*_{1}*g*_{3})*, S*_{2} =*S*(*f*_{2}*g*_{2}*, f*_{1}*g*_{1})*, S*_{3} =*S*(*f*_{2}*g*_{3}*, f*_{1}*g*_{2})*}*with *δ*(*G*_{g}) =
*n*_{1} +*n*_{2} + 2. For *G*_{g,}_{1} = *G*_{g} *∪ {r*_{1}*, r*_{2}*, r*_{3}*}*, *δ*(*G*_{g,}_{1}) is either 5 or 6. If *δ*(*G*_{g,}_{1}) = 5, then
*G*_{g,}_{1} is a Groebner basis. If *δ*(*G*_{g,}_{1}) = 6, then there is exactly one nonzero polynomial
in *{r*_{1}*, r*_{2}*, r*_{3}*}*. For *r*_{i} = 0, LM(*r*_{i}) is *XY*, *Y*^{2} or *X*^{3} and it is enough to consider of the
following S-polynomials:

(i) *S*(*f*_{1}*g*_{1}*, r*_{i}) and*S*(*f*_{1}*g*_{2}*, r*_{i}) if LM(*r*_{i}) =*XY*;

(ii) *S*(*f*_{1}*g*_{2}*, r*_{i}) and *S*(*f*_{1}*g*_{3}*, r*_{i}) if LM(*r*_{i}) =*Y*^{2};
(iii) *S*(*f*_{2}*g*_{1}*, r*_{i}) and *S*(*f*_{2}*g*_{2}*, r*_{i}) if LM(*r*_{i}) =*X*^{3}.

For a nonzero remainder *r* of these S-polynomials on division by *G*_{g,}_{1}, *G*_{g,}_{1} *∪ {r}* is a
Groebner basis for*ϕ*^{−1}(*I**D*_{1}+*D*_{2}) since*δ*(*G**g,*1) =*n*_{1}+*n*_{2}+ 1. Thus, the number of divisions
to be done for getting a Groebner basis is 5 at most.

Since *δ*(*H*) = 5 with ∆(*H*) *⊂ {*1*, X, Y, X*^{2}*, XY, Y*^{2}*, X*^{3}*}*, we have the following di-
agrams on LM(*H*) with the same result on *G* as that of **1***.* **LM**(**G**_{1}) =*{***X***,***Y**^{2}*}* in
**V***.* **n**_{1} =**2***,***n**_{2} =**3** except (2).

b b r r r b b r r r b r r r r r r r r r

*i*
*j*

(1)

LM(*H*)

=*{X*^{2}*, XY*^{2}*, Y*^{3}*}*

*⇔*LM(*G*)

=*{X, Y*^{2}*}*

b b b b r b r r r r r r r r r r r r r r

*i*
*j*

(2)

LM(*H*)

=*{XY, Y*^{2}*, X*^{4}*}*

*⇔*LM(*G*)

=*{X, Y}*

b b b r r b r r r r b r r r r r r r r r

*i*
*j*

(3)

LM(*H*)

=*{XY, X*^{3}*, Y*^{3}*}*

*⇔*LM(*G*)

=*{Y, X*^{2}*}*

b b b r r b b r r r r r r r r r r r r r

*i*
*j*

(4)

LM(*H*)

=*{Y*^{2}*, X*^{3}*, X*^{2}*Y}*

*⇔*LM(*G*)

=*{X*^{2}*, XY, Y*^{2}*}*
In the case of (2), *n*^{} = 2. Thus LM(*v*_{1}) is either *X* or *Y*. Let *H* = *{h*_{1}*, h*_{2}*, h*_{3}*}* with
LM(*h*_{1}) = *XY,*LM(*h*_{2}) = *Y*^{2}*,*LM(*h*_{3}) = *X*^{4}. Then *{h*_{1}*, F, XF* *−Y*^{2}*h*_{1}*}* is a Groebner
basis for *h*_{1}*, F*.

Now, consider on LM(*v*_{1}). Suppose that LM(*v*_{1}) = *Y*. Then LM(*G*) =*{Y, X*^{2}*}*. Since
*v*_{1}*h*_{2} *∈ h*_{1}*, F*,

*v*_{1}*h*_{2} =*q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}*F* +*q*_{2}_{,}_{3}(*XF* *−Y*^{2}*h*_{1})

for *q*_{2}_{,}_{1}*, q*_{2}_{,}_{2}*, q*_{2}_{,}_{3} *∈* *K*[*X, Y*] with LM(*q*_{2}_{,}_{1}) *≤* *Y, q*_{2}_{,}_{2} = 1*, q*_{2}_{,}_{3} = 0. Then *q*_{2}_{,}_{1} = *kv*_{1} for
*k* *∈* *K* since *q*_{2}_{,}_{1} *∈* *I* with LM(*q*_{2}_{,}_{1}) *≤* *Y*. It follows that *v*_{1}(*h*_{2} *−kh*_{1}) = *F* . It is a
contradiction since *F* is irreducible. Hence we have LM(*v*_{1}) =*X* and LM(*G*) =*{X, Y}*.

Since *v*_{1}*h*_{2} *∈ h*_{1}*, F*, we have

*v*_{1}*h*_{2} =*q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}*F* +*q*_{2}_{,}_{3}(*XF* *−Y*^{2}*h*_{1})

for *q*_{2}*,*1*, q*_{2}*,*2*, q*_{2}*,*3 *∈K*[*X, Y*] with LT(*q*_{2}*,*1) =*Y, q*_{2}*,*2 =*q*_{2}*,*3 = 0. It follows that *{v*_{1}*, q*_{2}*,*1*}*is a
Groebner basis for *I*. Thus *G*=*{v*_{1}*, v*_{2}*}* for the remainder *v*_{2} of *q*_{2}_{,}_{1} on division by *v*_{1}.

**VI***.* **n**_{1} =**3***,***n**_{2} =**3**

If the reduced Groebner bases are

*G*_{1} = *{f*_{1}(*X, Y*) = *X*^{2}+*A*_{1}*Y* +*B*_{1}*X*+*C*_{1}*, f*_{2}(*X, Y*) = *XY* +*A*_{2}*Y* +*B*_{2}*X*+*C*_{2}*,*
*f*_{3}(*X, Y*) = *Y*^{2}+*A*_{3}*Y* +*B*_{3}*X*+*C*_{3}*},*

*G*_{2} = *{g*_{1}(*X, Y*) =*X*^{2} +*a*_{1}*Y* +*b*_{1}*X*+*c*_{1}*, g*_{2}(*X, Y*) = *XY* +*a*_{2}*Y* +*b*_{2}*X*+*c*_{2}*,*
*g*_{3}(*X, Y*) =*Y*^{2}+*a*_{3}*Y* +*b*_{3}*X*+*c*_{3}*},*

we have the following diagram on LM(*G*_{g}).

e e e e u

e e e u u

e e u u u

u u u u u

u u u u u

0 1 2 3 4 *i*

1
2
3
4
*j*

LM(*G*_{g}) =*{X*^{4}*, Y*^{3}*, X*^{3}*Y, X*^{2}*Y*^{2}*, XY*^{3}*, Y*^{4}*}*
(*i, j*)*↔X*^{i}*Y*^{j}

MD = (0*,*4) : *f*_{3}*g*_{3}

(0*,*3) : *F* (1*,*3) :*f*_{2}*g*_{3}*, f*_{3}*g*_{2}
(2*,*2) : *f*_{1}*g*_{3}*, f*_{2}*g*_{2}*, f*_{3}*g*_{1}

(3*,*1) : *f*_{1}*g*_{2}*, f*_{2}*g*_{1}
(4*,*0) : *f*_{1}*g*_{1}

It follows that *S* = *{S*_{1} = *S*(*f*_{1}*g*_{2}*, f*_{2}*g*_{1})*, S*_{2} = *S*(*f*_{1}*g*_{3}*, f*_{2}*g*_{2})*, S*_{3} = *S*(*f*_{1}*g*_{3}*, f*_{3}*g*_{1})*, S*_{4} =
*S*(*f*_{2}*g*_{3}*, f*_{3}*g*_{2})*, S*_{5} = *S*(*f*_{2}*g*_{3}*, F*)*, S*_{6} = *S*(*f*_{3}*g*_{3}*, F*)*}* with *δ*(*G*_{g}) = *n*_{1} +*n*_{2} + 3. For *G*_{g,}_{1} =
*G*_{g}*∪{r*_{1}*, r*_{2}*, r*_{3}*, r*_{4}*, r*_{5}*, r*_{6}*}*,*δ*(*G*_{g,}_{1}) is 6, 7 or 8. If*δ*(*G*_{g,}_{1}) = 6, then*G*_{g,}_{1} is a Groebner basis.

If*δ*(*G*_{g,}_{1}) = 7, then there are one or two nonzero polynomials in *{r*_{1}*, r*_{2}*, r*_{3}*, r*_{4}*, r*_{5}*, r*_{6}*}*. For
all *r*_{i} = 0 in *{r*_{1}*, r*_{2}*, r*_{3}*, r*_{4}*, r*_{5}*, r*_{6}*}*, we compute the remainders of S-polynomials *S*(*r*_{i}*, f*)
and *S*(*r*_{i}*, g*) for a nearest polynomial *f* to *r*_{i} in the lower right-hand and a nearest
polynomial *g* to *r*_{i} in the upper left-hand of *G*_{g,}_{1} as considering the leading monomi-
als. Then, for a nonzero remainder *r* of S-polynomials, *G*_{g,}_{1} *∪ {r}* is a Groebner basis
since *δ*(*G*_{g,}_{1}) = *n*_{1} +*n*_{2} + 1. If *δ*(*G*_{g,}_{1}) = 8, then there is exactly one nonzero poly-
nomial in *{r*_{1}*, r*_{2}*, r*_{3}*, r*_{4}*, r*_{5}*, r*_{6}*}*. For a nonzero *r*_{i} in *{r*_{1}*, r*_{2}*, r*_{3}*, r*_{4}*, r*_{5}*, r*_{6}*}*, it is enough to
consider the S-polynomials *S*(*r*_{i}*, f*) and *S*(*r*_{i}*, g*) for a nearest polynomial *f* to *r*_{i} in the
lower right-hand and a nearest polynomial *g* to*r*_{i} in the upper left-hand of*G*_{g,}_{1}. Let *r*_{i,}_{1}
be the remainder of *S*(*r*_{i}*, f*) on division by *G*_{g,}_{1} and let *r*_{i,}_{2} be the remainder of *S*(*r*_{i}*, g*)
on division by *G*_{g,}_{1} *∪ {r*_{i,}_{1}*}*. For *G*_{g,}_{2} = *G*_{g,}_{1}*∪ {r*_{i,}_{1}*, r*_{i,}_{2}*}*, deg(*G*_{g,}_{2}) is either 6 or 7. If
*δ*(*G*_{g,}_{2}) = 6, then *G*_{g,}_{2} is a Groebner basis. If *δ*(*G*_{g,}_{2}) = 7, then there is only one nonzero
polynomial in *{r*_{i,}_{1}*, r*_{i,}_{2}*}*. For a nonzero *r*_{i,j} in *{r*_{i,}_{1}*, r*_{i,}_{2}*}*, it is enough to consider the

S-polynomials *S*(*r*_{i,j}*, f*) and*S*(*r*_{i,j}*, g*) for a nearest polynomial *f* to*r*_{i,j} in the lower right-
hand and a nearest polynomial *g* to *r*_{i,j} in the upper left-hand of *G*_{g,}_{2}. For a nonzero
remainder *r* of those S-polynomials on division by *G*_{g,}_{2}, *G*_{g,}_{2} *∪ {r}* is a Groebner basis
since *δ*(*G**g,*2) = *n*_{1} +*n*_{2} + 1. Thus, the number of divisions to be done for getting a
Groebner basis is 10 at most. In particular, if *G*_{1} = *G*_{2}, the number of divisions to be
done is 7 at most.

Since *δ*(*H*) = 6 with ∆(*H*) *⊂ {*1*, X, Y, X*^{2}*, XY, Y*^{2}*, X*^{3}*, X*^{2}*Y, XY*^{2}*}*, we have the fol-
lowing diagrams on LM(*H*), which are followed by LM(*G*).

b b r r r b b r r r b b r r r r r r r r r r r r r

*i*
*j*

(1)

LM(*H*)

=*{**X*^{2}*,Y*^{3}*}*

*⇔*LM(*G*)

=*{*1*}*

b b b b r b r r r r b r r r r r r r r r r r r r r

*i*
*j*

(2)

LM(*H*)

=*{**XY,X*^{4}*,Y*^{3}*}*

*⇔*LM(*G*)

=*{**X,Y*^{2}*}*

b b b r r b b b r r r r r r r r r r r r r r r r r

*i*
*j*

(3)

LM(*H*)

=*{**Y*^{2}*,X*^{3}*}*

*⇔*LM(*G*)

=*{**X,Y**}*

b b b b r b b r r r r r r r r r r r r r r r r r r

*i*
*j*

(4)

LM(*H*)

=*{**Y*^{2}*,X*^{2}*Y,X*^{4}*}*

*⇔*LM(*G*)

=*{**Y,X*^{2}*}*

b b b r r b b r r r b r r r r r r r r r r r r r r

*i*
*j*

(5)

LM(*H*)

=*{**X*^{3}*,X*^{2}*Y,XY*^{2}*,Y*^{3}*}*

*⇔*LM(*G*)

=*{**X*^{2}*,XY,Y*^{2}*}*

In the case of (1), *n*^{} = 0. Thus*n* = 0 and*G*=*{*1*}*.

In the case of (2), *n*^{} = 1. Thus LM(*v*_{1}) = *X* and LM(*G*) = *{X, Y*^{2}*}*. Let *H* =
*{h*_{1}*, h*_{2}*, h*_{3}*}* with LM(*h*_{1}) = *XY,*LM(*h*_{2}) =*X*^{4}*,*LM(*h*_{3}) = *Y*^{3}. Then *{h*_{1}*, F, XF* *−Y*^{2}*h*_{1}*}*
is a Groebner basis for *h*_{1}*, F*. Since*v*_{1}*h*_{2} *∈ h*_{1}*, F*, we have

*v*_{1}*h*_{2} =*q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}*F* +*q*_{2}_{,}_{3}(*XF* *−Y*^{2}*h*_{1})

for *q*_{2}_{,}_{1}*, q*_{2}_{,}_{2}*, q*_{2}_{,}_{3} *∈* *K*[*X, Y*] with LM(*q*_{2}_{,}_{1}) *≤* *XY,*LM(*q*_{2}_{,}_{2}) *≤* 1*, q*_{2}_{,}_{3} = 1. It follows that
*{v*_{1}*, Y*^{2}*−q*_{2}_{,}_{1}*}*is a Groebner basis for*I*. Thus*G*=*{v*_{1}*, v*_{2}*}*for the remainder*v*_{2} of*Y*^{2}*−q*_{2}_{,}_{1}
on division by *v*_{1}.

In the case of (3), *n*^{} = 2. Thus LM(*v*_{1}) is either *X* or *Y*. Let *H* = *{h*_{1}*, h*_{2}*}* with
LM(*h*_{1}) =*Y*^{2}*,*LM(*h*_{2}) =*X*^{3}. Then *{h*_{1}*, F* *−Y h*_{1}*}* is a Groebner basis for *h*_{1}*, F*. Since
LM(*v*_{1}*h*_{i}) *∈ Y*^{2}*, X*^{4} for all *h*_{i} *∈* *H*, we have LM(*v*_{1}) = *X*. It follows that LM(*G*) =
*{X, Y}*. Since*v*_{1}*h*_{2} *∈ h*_{1}*, F*, we have

*v*_{1}*h*_{2} =*q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}(*F* *−Y h*_{1})

for *q*_{2}_{,}_{1}*, q*_{2}_{,}_{2} *∈* *K*[*X, Y*] with LM(*q*_{2}_{,}_{1}) *≤* *X, q*_{2}_{,}_{2} = 1. It follows that *{v*_{1}*, Y* *−q*_{2}_{,}_{1}*}* is a
Groebner basis for *I*. Thus *G* = *{v*_{1}*, v*_{2}*}* for the remainder *v*_{2} of *Y* *−q*_{2}_{,}_{1} on division by
*v*_{1}.

In the case of (4), *n*^{} = 2. Thus LM(*v*_{1}) is either *X* or *Y*. Let *H* = *{h*_{1}*, h*_{2}*, h*_{3}*}*
with LM(*h*_{1}) = *Y*^{2}*,*LM(*h*_{2}) = *X*^{2}*Y,*LM(*h*_{3}) = *X*^{4}. Then *{h*_{1}*, F* *−Y h*_{1}*}* is a Groebner
basis for *h*_{1}*, F*. Since LM(*v*_{1}*h*_{2}) *∈ Y*^{2}*, X*^{4}, we have LM(*v*_{1}) = *Y*. It follows that
LM(*G*) =*{Y, X*^{2}*}*. Since *v*_{1}*h*_{2} *∈ h*_{1}*, F*, we have

*v*_{1}*h*_{2} =*q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}(*F* *−Y h*_{1})

for *q*_{2}_{,}_{1}*, q*_{2}_{,}_{2} *∈K*[*X, Y*] with LT(*q*_{2}_{,}_{1}) =*X*^{2}*,*LM(*q*_{2}_{,}_{2})*≤*1. It follows that*{v*_{1}*, q*_{2}_{,}_{1}*−Y q*_{2}_{,}_{2}*}*
is a Groebner basis for*I*. Thus*G*=*{v*_{1}*, v*_{2}*}*for the remainder*v*_{2} of*q*_{2}_{,}_{1}*−Y q*_{2}_{,}_{2} on division
by*v*_{1}.

In the case of (5), *n*^{} = 3 and LM(*v*_{1}) =*X*^{2}. Further, LM(*G*) = *{X*^{2}*, XY, Y*^{2}*}*. Let
*H* = *{h*_{1}*, h*_{2}*, h*_{3}*, h*_{4}*}* with LM(*h*_{1}) = *X*^{3}*,*LM(*h*_{2}) = *X*^{2}*Y,*LM(*h*_{3}) = *XY*^{2}*,*LM(*h*_{4}) = *Y*^{3}.
Then*{h*_{1}*, F}* is a Groebner basis for*h*_{1}*, F*. Since*v*_{1}*h*_{i} *∈ h*_{1}*, F* for all*h*_{i} *∈H*, we have

*v*_{1}*h*_{2} =*q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}*F*

for *q*_{2}_{,}_{1}*, q*_{2}_{,}_{2} *∈K*[*X, Y*] with LT(*q*_{2}_{,}_{1}) =*XY,*LM(*q*_{2}_{,}_{2})*≤X*, and
*v*_{1}*h*_{3} =*q*_{3}_{,}_{1}*h*_{1}+*q*_{3}_{,}_{2}*F*

for *q*_{3}_{,}_{1}*, q*_{3}_{,}_{2} *∈K*[*X, Y*] with LT(*q*_{3}_{,}_{1}) = *Y*^{2}*,*LM(*q*_{3}_{,}_{2})*≤* *Y*. It follows that *{v*_{1}*, q*_{2}_{,}_{1}*, q*_{3}_{,}_{1}*}* is
a Groebner basis for *I*. Thus *G*= *{v*_{1}*, v*_{2}*, v*_{3}*}* for the remainder *v*_{2} of *q*_{2}_{,}_{1} on division by
*v*_{1} and the remainder *v*_{3} of *q*_{3}_{,}_{1} on division by *{v*_{1}*, v*_{2}*}*.

**Chapter 5** **Appendix**

In this appendix, we consider on the sum of normal divisors*D*_{1} and*D*_{2} of a*C*_{34}curve
by using the relation between the reduced Groebner basis for *ϕ*^{−1}(*I*_{D}_{1}) and the reduced
Groebner basis for *ϕ*^{−1}(*I**D*_{2}). In particular, we consider on the reduced Groebner basis
*H* for *ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2}). Let *D* be the normal divisor such that *D* *∼D*_{1}+*D*_{2}. For the given
*H*, the computation of the reduced Groebner basis *G*for *ϕ*^{−1}(*I**D*) is presented in Section
4.4. Now that the reduced Groebner basis is easily computed by a Groebner basis ([5]),
we compute a Groebner basis. Here, we use the notation in Chapter 4. Sometimes we
represent a polynomial *f*(*X, Y*)*∈K*[*X, Y*] as *f*.

For the sum *D*_{1}+*D*_{2} with *n*_{1} = 1 and *n*_{2} = 1 or 2, the result on*G*is given in Section
4.4.

Now, we consider the sum *D*_{1} +*D*_{2} with *n*_{1} = 1 and *n*_{2} = 3. Let *D*_{1} = *P* *− ∞* and
*D*_{2} =*E*_{2}*−*3*· ∞* be normal divisors of*C* with

*G*_{1} =*{f*_{1}(*X, Y*) =*X*+*C*_{1}*, f*_{2}(*X, Y*) =*Y* +*C*_{2}*}*

and *G*_{2} =*{g*_{1}(*X, Y*)*, g*_{2}(*X, Y*)*, g*_{3}(*X, Y*)*}*, where

*g*_{1}(*X, Y*) = *X*^{2} +*a*_{1}*Y* +*b*_{1}*X* +*c*_{1}*,*
*g*_{2}(*X, Y*) = *XY* +*a*_{2}*Y* +*b*_{2}*X* +*c*_{2}*,*
*g*_{3}(*X, Y*) = *Y*^{2} +*a*_{3}*Y* +*b*_{3}*X* +*c*_{3}*.*

Let

*g*^{}_{2}(*X, Y*) = *XY* + (*−a*_{2}+*b*_{1})*Y* + (*a*^{2}_{1}*−a*_{3} *−a*_{1}*s*_{2})*X*

*−a*^{2}_{1}*a*_{2}+*a*_{2}*a*_{3}*−a*^{2}_{1}*b*_{1}*−a*_{3}*b*_{1}+*a*_{1}*b*_{3}*−a*_{1}*s*_{1}+*a*_{1}*a*_{2}*s*_{2}+*a*^{2}_{1}*t*_{3}*,*
*g*^{}_{3}(*X, Y*) = *Y*^{2}+ (*a*^{2}_{1} *−b*_{2}*−a*_{1}*s*_{2})*Y* + (2*a*_{1}*b*_{1}*−b*_{3}+*s*_{1}*−b*_{1}*s*_{2}*−a*_{1}*t*_{3})*X*

*−*2*a*_{1}*a*^{2}_{2}+ 2*a*^{2}_{1}*a*_{3}+ 2*a*_{1}*a*_{2}*b*_{1}*−a*_{1}*b*^{2}_{1}*−*3*a*^{2}_{1}*b*_{2}+*b*^{2}_{2}+*a*_{2}*b*_{3}*−b*_{1}*b*_{3}+*s*_{0}+*a*^{2}_{2}*s*_{2}

*−a*_{1}*a*_{3}*s*_{2}*−a*_{2}*b*_{1}*s*_{2}+ 2*a*_{1}*b*_{2}*s*_{2} *−a*_{1}*t*_{2}+*a*_{1}*b*_{1}*t*_{3}*.*

Then *{g*_{1}(*X, Y*)*, g*_{2}^{}(*X, Y*)*, g*_{3}^{}(*X, Y*)*}* is the reduced Groebner basis for the normal ideal
*ϕ*^{−1}(*I*_{D}

2), where *D*_{2}^{} is the normal divisor such that *D*_{2}^{} *∼ −D*_{2}.

For *S* =*{S*_{1} =*S*(*f*_{1}*g*_{2}*, f*_{2}*g*_{1})*, S*_{2} =*S*(*f*_{1}*g*_{3}*, f*_{2}*g*_{2})*, S*_{3} = (*F, f*_{2}*g*_{3})*}* and the remainder *r*_{3}
of *S*_{3} on division by *G*_{g} =*{f*_{i}*g*_{j} *|f*_{i} *∈G*_{1}*, g*_{j} *∈G*_{2}*} ∪ {F}*, we have

*S*_{1} = *−a*_{1}*Y*^{2}+ (*a*_{2}*−b*_{1}+*C*_{1})*XY* +*· · ·,*

*S*_{2} = (*−a*_{2}+*C*_{1})*Y*^{2}+ (*a*_{3}*−b*_{2}*−C*_{2})*XY* +*· · ·,*

*r*_{3} = *−*(*a*_{3}+*C*_{2})*Y*^{2}+ (*a*_{1}*a*_{2}+*a*_{1}*b*_{1}*−b*_{3}+*a*_{1}*C*_{1}+*s*_{1}*−a*_{2}*s*_{2}*−C*_{1}*s*_{2}*−a*_{1}*t*_{3})*XY* +*· · ·.*
Here,

*g*_{1}(*−C*_{1}*,−C*_{2}) = (*a*_{2}*−b*_{1} +*C*_{1})(*−a*_{2}+*C*_{1}) +*a*_{1}(*a*_{3}*−b*_{2}*−C*_{2})*,*

*g*_{2}^{}(*−C*_{1}*,−C*_{2}) = *−a*_{1}(*a*_{1}*a*_{2}+*a*_{1}*b*_{1}*−b*_{3}+*a*_{1}*C*_{1}+*s*_{1}*−a*_{2}*s*_{2}*−C*_{1}*s*_{2}*−a*_{1}*t*_{3})
+(*a*_{2}*−b*_{1}+*C*_{1})(*a*_{3}+*C*_{2})

by Theorem 4.2.2. It follows that:

(a) *Y*^{2}*, XY* *∈*LM(*S*_{1}*, S*_{2}) if *g*_{1}(*−C*_{1}*,−C*_{2})= 0;

(b) *Y*^{2}*, XY* *∈*LM(*S*_{1}*, r*_{3}) if *g*_{2}^{}(*−C*_{1}*,−C*_{2})= 0.

Since *δ*(*H*) = 4 with ∆(*H*) *⊂ {*1*, X, Y, X*^{2}*, XY, Y*^{2}*}*, LM(*H*) is one of the following:

*{X*^{2}*, XY, Y*^{3}*}*; *{X*^{2}*, Y*^{2}*}*; and *{XY, Y*^{2}*, X*^{3}*}*. For every *h*(*X, Y*) *∈* *H*, *h*(*X, Y*) is divis-
ible by *G*_{2} since *ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2}) *⊂* *ϕ*^{−1}(*I*_{D}_{2}). It implies that *X*^{2} *∈* LM(*H*) if and only if
*g*_{1}(*X, Y*)*∈H*. In other words,*X*^{2} *∈*LM(*H*) if and only if (*D*^{}_{2})^{+}*≥P* = (*−C*_{1}*,−C*_{2}), i.e.

*g*_{1}(*−C*_{1}*,−C*_{2}) =*g*_{2}^{}(*−C*_{1}*,−C*_{2}) = *g*_{3}^{}(*−C*_{1}*,−C*_{2}) = 0.

As a result, we have the following on the ideal *ϕ*^{−1}(*I*_{D}_{1}_{+}_{D}_{2})*⊂K*[*X, Y*].

(i) If *g*_{1}(*−C*_{1}*,−C*_{2})= 0, then we have a Groebner basis

*{g*_{2}+*k*_{1}*g*_{1}*, g*_{3}+*k*_{2}*g*_{1}*, f*_{1}*g*_{1}*}*with LM(*H*) = *{XY, Y*^{2}*, X*^{3}*},*

where *k*_{1} =*−g*_{1}(*−C*_{1}*,−C*_{2})^{−1}*g*_{2}(*−C*_{1}*,−C*_{2})*, k*_{2} =*−g*_{1}(*−C*_{1}*,−C*_{2})^{−1}*g*_{3}(*−C*_{1}*,−C*_{2})*∈K*.

(ii) If *g*_{1}(*−C*_{1}*,−C*_{2}) = 0 and *g*_{2}^{}(*−C*_{1}*,−C*_{2})= 0, then we have a Groebner basis
*{S*_{1}*, r*_{3}*, S*(*S*_{1}*, r*_{3})*, f*_{1}*g*_{1}*}* with LM(*H*) =*{XY, Y*^{2}*, X*^{3}*}.*

(iii) If *g*_{1}(*−C*_{1}*,−C*_{2}) =*g*^{}_{2}(*−C*_{1}*,−C*_{2}) = 0 and*g*^{}_{3}(*−C*_{1}*,−C*_{2})= 0, then *a*_{1} = 0 and we
have a Groebner basis

*{S*_{2}*, r*_{3}*, g*_{2}+*F*_{Y}(*−C*_{1}*,−C*_{2})^{−1}*F*_{X}(*−C*_{1}*,−C*_{2})*g*_{1}*, f*_{1}*g*_{1}*}*with LM(*H*) =*{XY, Y*^{2}*, X*^{3}*}.*
(iv) If *g*_{1}(*−C*_{1}*,−C*_{2}) = *g*_{2}^{}(*−C*_{1}*,−C*_{2}) = *g*_{3}^{}(*−C*_{1}*,−C*_{2}) = 0, then we have a Groebner
basis

*{g*_{1}*, g*_{2}*, f*_{2}*g*_{3}*}*with LM(*H*) = *{X*^{2}*, XY, Y*^{3}*}*
in the case of *a*_{1} =*−a*_{2}+*C*_{1} =*a*_{3}+*C*_{2} = 0, and

*{S*_{1}*, S*_{2}*, r*_{3}*, g*_{1}*}* with LM(*H*) =*{X*^{2}*, Y*^{2}*}*
in the other cases.

Hence we have *G* when*n*_{1} = 1.

In the case of *n*_{1} = 2 and *n*_{2} = 2 or 3, the zero divisor *E*_{1} = *P*_{1} +*P*_{2} is obtained by
the equations *f*_{1}(*X, Y*) = 0 and *f*_{2}(*X, Y*) = 0. Thus *G* can be obtained by the sum of
*P*_{1}*− ∞* and (*P*_{2}*− ∞*) +*D*_{2}.

Now, we consider the last case *n*_{1} =*n*_{2} = 3. Let *E*_{1} =*P*_{1}+*P*_{2}+*P*_{3} with*P**i* = (*x**i**, y**i*).

Then

(*X*+*A*_{2})*f*_{2}(*X, Y*)*−A*_{1}*f*_{1}(*X, Y*) =

3
*i*=1

(*X−x*_{i})*.* (1)

At ﬁrst, we compute the reduced Groebner basis *G*_{c} for
*f*_{1}*, f*_{2}*, f*_{3}*, f*_{1}*−g*_{1}*, f*_{2}*−g*_{2}*, f*_{3}*−g*_{3}*.*
Then LM(*G**c*) =*{*1*},{X, Y},{X, Y*^{2}*},{Y, X*^{2}*}*or *{X*^{2}*, XY, Y*^{2}*}*.

If LM(*G*_{c}) = *{*1*}*, then*h*(*X, Y*)*∈ϕ*^{−1}(*L*(*∞ · ∞ −*(*E*_{1}+*E*_{2}))) if and only if *h*(*X, Y*) is
divisible by *G*_{1} and by *G*_{2}. Thus the reduced Groebner basis *H* is easily computed. For
example, if *f*_{1}(*X, Y*) = *g*_{1}(*X, Y*), then *h*_{1}(*X, Y*) =*f*_{1}(*X, Y*)*∈H*.

If LM(*G*_{c}) = *{X, Y},{X, Y*^{2}*}* or *{Y, X*^{2}*}*, then *E*_{1} =*P*_{1}+*P*_{2}+*P*_{3} is obtained by *G*_{c}
and (1). Thus *G* can be obtained by (*P*_{1}*− ∞*) + ((*P*_{2}*− ∞*) + ((*P*_{3}*− ∞*) +*D*_{2}))*.*

If LM(*G*_{c}) = *{X*^{2}*, XY, Y*^{2}*}*, then *D*_{1} = *D*_{2}. In the case of *A*_{1} = 0, *E*_{1} = (*A*_{2} *−*
*B*_{1}*,−B*_{2}) + (*−A*_{2}*, β*_{1}) + (*−A*_{2}*, β*_{2}) for the roots *β*_{1}*, β*_{2} of*Y*^{2}+*A*_{3}*Y* *−B*_{3}*A*_{2}+*C*_{3} = 0. Thus
*G*can be obtained by (*A*_{2}*−B*_{1}*,−B*_{2})*− ∞*+ ((*−A*_{2}*, β*_{1})*− ∞*+ ((*−A*_{2}*, β*_{2})*− ∞*+*D*_{2}))*.*In
the case of *A*_{1} = 0, to avoid using a cubic equation, we use the method given in Section
4.4 with the S-polynomials whose number is 7 at most.

From now on, we consider the general case *f*_{1}*, f*_{2}*, f*_{3}*, f*_{1} *−g*_{1}*, f*_{2} *−g*_{2}*, f*_{3}*−g*_{3} = 1.
Let

*M*(*G*_{1}*, G*_{2}) =

*A*_{1}*−a*_{1} *A*_{2}*−a*_{2} *A*_{3}*−a*_{3}
*B*_{1}*−b*_{1} *B*_{2}*−b*_{2} *B*_{3}*−b*_{3}
*C*_{1}*−c*_{1} *C*_{2}*−c*_{2} *C*_{3}*−c*_{3}

*.*

Assume that det*M*(*G*_{1}*, G*_{2})= 0. Then the elements of *H* are
*h*_{1}(*X, Y*) = (*X*+*k*_{1})*f*_{1}+*k*_{2}*f*_{2}+*k*_{3}*f*_{3}*,*
*h*_{2}(*X, Y*) = *k*_{4}*f*_{1}+ (*X*+*k*_{5})*f*_{2}+*k*_{6}*f*_{3}*,*
*h*_{3}(*X, Y*) = *k*_{7}*f*_{1}+*k*_{8}*f*_{2}+ (*X*+*k*_{9})*f*_{3}

and the remainder of *F*(*X, Y*) on division by *{h*_{1}(*X, Y*)*, h*_{2}(*X, Y*)*, h*_{3}(*X, Y*)*}* for *k**i* *∈* *K*
such that

*k*_{1}
*k*_{2}
*k*_{3}

= *M*(*G*_{1}*, G*_{2})^{−1}

*a*_{2}(*A*_{1}*−a*_{1}) +*a*_{1}(*B*_{1}*−b*_{1})

*b*_{2}(*A*_{1}*−a*_{1}) +*b*_{1}(*B*_{1}*−b*_{1})*−*(*C*_{1}*−c*_{1})
*c*_{2}(*A*_{1}*−a*_{1}) +*c*_{1}(*B*_{1} *−b*_{1})

*,*

*k*_{4}
*k*_{5}
*k*_{6}

= *M*(*G*_{1}*, G*_{2})^{−1}

*a*_{2}(*A*_{2}*−a*_{2}) +*a*_{1}(*B*_{2}*−b*_{2})

*b*_{2}(*A*_{2}*−a*_{2}) +*b*_{1}(*B*_{2}*−b*_{2})*−*(*C*_{2}*−c*_{2})
*c*_{2}(*A*_{2}*−a*_{2}) +*c*_{1}(*B*_{2} *−b*_{2})

*,*

*k*_{7}
*k*_{8}
*k*_{9}

= *M*(*G*_{1}*, G*_{2})^{−1}

*a*_{2}(*A*_{3}*−a*_{3}) +*a*_{1}(*B*_{3}*−b*_{3})

*b*_{2}(*A*_{3}*−a*_{3}) +*b*_{1}(*B*_{3}*−b*_{3})*−*(*C*_{3}*−c*_{3})
*c*_{2}(*A*_{3}*−a*_{3}) +*c*_{1}(*B*_{3} *−b*_{3})

*.*

Since*D*^{} = (*h*_{1})*−*(*D*_{1}+*D*_{2}) is a normal divisor with the pole degree 3, we have a unique
polynomial *v*_{1} *∈* *K*[*X, Y*] with LT(*v*_{1}) = *X*^{2} such that *v*_{1}*h*_{i} *∈ h*_{1}*, F* for all *h*_{i} *∈* *H*.

*{h*_{1}*, F}* is a Groebner basis for *h*_{1}*, F*since lcm(LM(*h*_{1})*,*LM(*F*)) = LM(*h*_{1})LM(*F*). For
the polynomials *v*_{1}*, q*_{2}_{,}_{1}*, q*_{2}_{,}_{2}*, q*_{3}_{,}_{1}*, q*_{3}_{,}_{2} *∈K*[*X, Y*] such that

*v*_{1}*h*_{2} = *q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}*F,*
*v*_{1}*h*_{3} = *q*_{3}_{,}_{1}*h*_{1}+*q*_{3}_{,}_{2}*F*

with LT(*v*_{1}) = *X*^{2}*,*LT(*q*_{2}_{,}_{1}) = *XY,*LM(*q*_{2}_{,}_{2}) *≤* *X,*LT(*q*_{3}_{,}_{1}) =*Y*^{2}*,*LM(*q*_{3}_{,}_{2}) *≤* *Y*, we have
*G*=*{v*_{1}*, v*_{2} =*q*_{2}_{,}_{1}*−c*_{2}_{,}_{1}*v*_{1}*, v*_{3} =*q*_{3}_{,}_{1}*−c*_{3}_{,}_{2}*v*_{2}*−c*_{3}_{,}_{1}*v*_{1}*}*, where *c*_{2}_{,}_{1} is the coeﬃcient of *X*^{2} in
*q*_{2}_{,}_{1} and *c*_{3}_{,}_{1}*, c*_{3}_{,}_{2} are the coeﬃcients of*X*^{2}*, XY* in*q*_{3}_{,}_{1}, respectively.

**Example**

Let *C* be a*C*_{34} curve deﬁned over *F*_{11} =*Z/*11*Z* by
*F*(*X, Y*) = *Y*^{3}+*X*^{4} + 1*.*

Let*D*_{1}*, D*_{2}be the normal divisors with the reduced Groebner bases*G*_{1}*, G*_{2}for their normal
ideals, respectively:

*G*_{1} =*{f*_{1} =*X*^{2}+ 8*Y* + 9*X*+ 9*, f*_{2} =*XY* + 4*Y* + 9*X*+ 8*, f*_{3} =*Y*^{2}+ 9*Y* + 9*X*+ 1*},*
*G*_{2} =*{g*_{1} =*X*^{2}+ 10*Y* + 7*X*+ 7*, g*_{2} =*XY* + 2*Y* + 4*X*+ 6*, g*_{3} =*Y*^{2}+ 7*Y* + 9*X*+ 2*}.*
Then *f*_{1}*, f*_{2}*, f*_{3}*, f*_{1} *−g*_{1}*, f*_{2} *−g*_{2}*, f*_{3} *−g*_{3}=1. For *G*_{1} and *G*_{2}, we have

*M*(*G*_{1}*, G*_{2}) =

9 2 2 2 5 0 2 2 10

with det*M*(*G*_{1}*, G*_{2})= 0*.*

Thus the elements of *H* are

*h*_{1}(*X, Y*) = (*X*+*k*_{1})*f*_{1}+*k*_{2}*f*_{2}+*k*_{3}*f*_{3}*,*
*h*_{2}(*X, Y*) = *k*_{4}*f*_{1}+ (*X*+*k*_{5})*f*_{2}+*k*_{6}*f*_{3}*,*
*h*_{3}(*X, Y*) = *k*_{7}*f*_{1}+*k*_{8}*f*_{2}+ (*X*+*k*_{9})*f*_{3}

and the remainder of*F*(*X, Y*) on division by*{h*_{1}(*X, Y*)*, h*_{2}(*X, Y*)*, h*_{3}(*X, Y*)*}*for

*k*_{1}
*k*_{2}
*k*_{3}

=

6 5 9

*,*

*k*_{4}
*k*_{5}
*k*_{6}

=

6 8 3

*,*

*k*_{7}
*k*_{8}
*k*_{9}

=

1 8 6

*.*

It follows that

*h*_{1} = *X*^{3}+ 9*Y*^{2}+ 2*XY* + 4*X*^{2}+ 6*Y* + 2*X*+ 4*,*
*h*_{2} = *X*^{2}*Y* + 3*Y*^{2} +*XY* + 4*X*^{2}+ 8*Y* + 7*X,*
*h*_{3} = *XY*^{2}+ 6*Y*^{2} + 6*XY* + 10*X*^{2}+ 6*Y* + 4*X*+ 2*.*

From

*v*_{1}*h*_{2} = *q*_{2}_{,}_{1}*h*_{1}+*q*_{2}_{,}_{2}*F,*
*v*_{1}*h*_{3} = *q*_{3}_{,}_{1}*h*_{1}+*q*_{3}_{,}_{2}*F*

with LM(*v*_{1}) =*X*^{2}*,*LT(*q*_{2}*,*1) =*XY,*LM(*q*_{2}*,*2)*≤X,*LT(*q*_{3}*,*1) = *Y*^{2}*,*LM(*q*_{3}*,*2)*≤X*, we have
*v*_{1} = *X*^{2}+ 3*Y* + 10*X*+ 10*,*

*q*_{2}_{,}_{1} = *XY* + 9*X*^{2}+ 3*X*+ 6*,*
*q*_{2}_{,}_{2} = 2*X*+ 9*,*

*q*_{3}_{,}_{1} = *Y*^{2}+ 9*XY* + 3*X*^{2}+ 8*Y* + 3*X*+ 9*,*
*q*_{3}_{,}_{2} = 2*Y* + 8*X*+ 6*.*

It follows that

*G*=*{v*_{1} =*X*^{2}+ 3*Y* + 10*X*+ 10*, v*_{2} =*XY* + 6*Y* +*X*+ 4*, v*_{3} =*Y*^{2} + 8*X*+ 9*}.*

**Bibliography**

[1] L. M. Adleman, J. DeMarrais and M. D. Huang, *A subexponential Algorithm for*
*Discrete Logarithms over the Rational Subgroup of the Jacobians of Large Genus*
*Hyperelliptic Curves over Finite Fields,* Algorithmic Number Theory, 28–40, Lecture
Notes in Comput. Sci. 877, Springer-Verlag, Berlin, 1994.

[2] S. Arita, *Algorithms for computation in Jacobian group of* *C*_{ab} *curve and their ap-*
*plication to discrete-log-based public key cryptosystems,* The mathematics of public
key cryptography, Fields Institute A. Odlyzko et al (eds.), 1999.

[3] D. G. Cantor, *Computing in the Jacobian of a hyperelliptic curve,* Math. Comp. 48
(1987), 95–101.

[4] D. G. Cantor, *On the analogue of the division polynomials for hyperelliptic curves,*
J. Reine Angew. Math. 447 (1994), 91–145.

[5] D. Cox, J. Little and D. O’shea, *Ideals, Varieties, and Algorithms,* Springer-Verlag,
Berlin, 1997.

[6] W. Fulton, *Algebraic Curves,* Benjamin, New York, 1969.

[7] S. D. Galbraith, S. M. Paulus and N. P. Smart, *Arithmetic on Superelliptic Curves,*
Math. Comp. 71 (2002), 393–405.

[8] V. D. Goppa,*Geometry and Codes,* Kluwer Academic Publishers, 1988.

[9] R. Harley,*adding.text,* http://cristal.inria.fr/˜harley/hyper/, 2000.

[10] R. Harley,*doubling.c,* http://cristal.inria.fr/˜harley/hyper/, 2000.

[11] R. Hartshorne, *Algebraic Geometry,* Springer-Verlag, 1977.

[12] M. -D. Huang and D. Ierardi, *Eﬃcient algorithms for the eﬀective Riemann-Roch*
*problem and for addition in the Jacobian of a curve,* J. Symbolic Comp. 18 (1994),
519–539.

[13] H. Kim, J. Cheon and S. Hahn, *Elliptic curve lifting problem and its applications,*
Proc. Japan Acad. 75 Ser. A (1999), 166–168.

[14] N. Koblitz, *Elliptic Curve Cryptosystems,* Math. Comp. 48 (1987), 203–209.

[15] N. Koblitz, *Hyperelliptic cryptosystems,* J. Cryptology 1 (1989), 139–150.

[16] N. Koblitz, *A course in number theory and cryptography,* 2nd ed., Grad. Texts in
Math. 114, Springer-Verlag, New York, 1994.

[17] N. Koblitz, *Algebraic Aspects of Cryptograph,* Springer-Verlag, 1998.

[18] R. Matsumoto, *The Cab curve - a generalization of the Weierstrass form to arbitrary*
*plane curves,* http://www.rmatsumoto.org/cab/html.

[19] K. McCurley, *The Discrete Logarithm Problem* (Boulder, CO, 1989), 49–74, Proc.

Sympos. Appl. Math. 42, Amer. Math. Soc., Proridence, RI, 1990.

[20] A. Menezes, *Elliptic Curve Public Key Cryptosystems,* Kluwer Acad. Publ., 1993.

[21] A. Menezes, T. Okamoto and S. A. Vanstone, *Reducing Elliptic Curve Logarithms*
*to Logarithms in a Finite Field,* IEEE Trans. Inform. Theory 39 (1993), 1639–1646.

[22] V. Miller, *Uses of elliptic curves in cryptography,* Advances in Cryptology-Proc.

Crypto ’85 (Santa Barbara, Calif., 1985), 417–426, Lecture Notes in Comput. Sci.

218, Springer-Verlag, New York, 1986.

[23] S. Miura, *Algebraic geometric codes on certain plane curves,* Trans. IEICE **J75-A**
(1992), 1735–1745 (Japanese).

[24] S. Miura, *Linear codes on aﬃne algebraic curves,* Trans. IEICE **J81-A** (1998),
1398–1421 (Japanese).

[25] Y. Morita, *Seisuron,* Tokyo Univ. Press, 1999 (Japanese).