2002/11/5(20:33)

Fujimoto, M. and Suzuki, M.
Osaka J. Math.
39 (2002), 1–23

CONSTRUCTION OF AFFINE PLANE CURVESWITH ONE PLACE AT INFINITY
MITSUSHI FUJIMOTO and MASAKAZU SUZUKI(Received January 12, 2001)

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1. Introduction

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COLet C be an irreducible algebraic curve in complex affine plane C2 . We say that
C has one place at infinity, if the closure of C intersects with the ∞-line in P2 atonly one point P and C is locally irreducible at that point P .
The problem of finding the canonical models of curves with one place at infinity under the polynomial transformations of the coordinates of C2 has been studied bymany mathematicians since Suzuki [17] and Abhyankar-Moh [2] proved independently
that the canonical model of C is a line when C is non-singular and simply connected.
Zaidenberg-Lin [19] proved that C has the canonical model of type y q = x p , where p
and q are coprime integers > 1, when C is singular and simply connected. A’CampoOka [5] studied the case of genus g ≤ 3 as an application of a resolution tower oftoric modifications. For the case g ≤ 4 Neumann [12] studied from the viewpointof the link at infinity, and Miyanishi [9] studied from the algebrico-geometric viewpoint. Nakazawa-Oka [11] gave the classifications of all the canonical models for thecase g ≤ 7 using the result of A’Campo-Oka, and gave the classifications for the caseg ≤ 16 without proof. Jaworski [8] studied normal forms of irreducible germs of functions of two variables with given Puiseux pairs. Oka [14, 15] gave the normal form ofplane curves which are locally irreducible at the origin and with a given sequence of
weight vectors corresponding to the Tschirnhausen-good resolution tower, and showedthat the moduli space of such curves is of the form (C∗ )a × Cb . Furthermore, Oka
translated this result to the case of affine curves with one place at infinity.
Also, Abhyankar-Moh [1, 3, 4] investigated properties of δ-sequences which aresequences of pole orders of approximate roots of C. This result is called AbhyankarMoh’s semigroup theorem. Sathaye-Stenerson [16] proved that if a sequence S of natural numbers satisfies Abhyankar-Moh’s condition then there exists a curve with one
place at infinity of the δ-sequence S. Suzuki [18] made it clear the relationship between the δ-sequence and the dual graph of the minimal resolution of the singularityof the curve C at infinity, and gave an algebrico-geometric proof of semigroup theorem and its inverse theorem due to Sathaye-Stenerson.
In this paper, we develop Suzuki’s result and give an algebrico-geometric proof of
Oka’s result (Theorem 7 and Corollary 1). We shall also give an algorithm to computeD

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M. FUJIMOTO AND M. SUZUKI

the normal form and the moduli space of the curve with one place at infinity from agiven δ-sequence1 .
Our construction method of normal forms is different from [8, 14, 15] in the following respects. First, this method uses δ-sequences generating semigroups of affine
plane curves with one place at infinity. Second, this method directly generates defining polynomials at the origin of curves with one place at infinity.
2.

Preparations

cαβ x α y β ,

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f (x, y) = (ux p + vy q )d +CO

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In this section, we introduce some definitions and facts which is needed to describe our theorem.
Let C be a curve with one place at infinity defined by a polynomial equation
f (x, y) = 0 in the complex affine plane C2 . Assume that degx f = m, degy f = n
and d = gcd(m, n). By the consideration of the Newton boundary, we can get

qα+pβ<pqd

where u, v ∈ C∗ , m = pd and n = qd. By a finitely many times of the coordinatetransformations of the form
x1 = xy1 = y + cx pONand the exchange of the coordinates x and y, we can reduce the polynomial f into
one of the following two types:(A) m = 1, n = 0(B) m = pd, n = qd, gcd(p, q) = 1, p > q > 1.
A curve of type (A) is a line. We call the curve of type (B) non-linearlizable. In thispaper, we shall consider only the curves of the type (B) from now on. The closure C
of C in the projective plane P2 passes through the intersection point O of the ∞-line
A and the line x = 0 by the assumption p > q.
Let us denote by E0 the (−1)-curve appeared by the blowing-up of the point O,
and continue to denote the proper transform of A by the same character A. Let a be
the natural number satisfying aq < p < (a + 1)q. If a = 1, then the proper transformof C is tangent to A, or else is tangent to E0 .

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1 The computer calculation by our algorithm verified the result of Nakazawa-Oka [11].

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CONSTRUCTION OF AFFINE PLANE CURVES

3

E0

C

y

blowing-up

A

C
A

y

x

x

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In case a > 1, after further a − 1 times of the blowing-ups of the point at infinityof the curve C, the proper transform of C is tangent to the (−1)-curve E1 obtainedby the last blowing-up. (In case a = 1, we set E1 = A.)
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E0
C

blowing-ups

y

A

x

Thus we get a compactification of C2 with the boundary curve of which the dualgraph is of the following form:
−a

−1

E0

E1

−2

−2

−2

−1

A

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By a − 1 times of the blowing-downs of the (−1)-curve on the right hand sidefrom A of the above dual graph, we get the following dual graph:
−a

0

E0

E1

.

Let (M1 , E0 ∪ E1 ) be the compactification of C2 thus obtained.
The intersection point of E0 and E1 is the indetermination point of f . Now, weblow up from the surface M1 the indetermination points of f successively, until the indetermination points of f disappear. Let Mf be the surface thus obtained. We denotethe proper transform in Mf of E0 (resp. E1 ) by the same character E0 (resp. E1 ). Let
Ei (2 ≤ i ≤ R) be the proper transform in Mf of the (−1)-curve obtained by the(i − 1)-th blowing-up. Furthermore, we set Ef = E0 ∪ E1 ∪ · · · ∪ ER .
The following theorem about the compactification (Mf , Ef ) of C2 is very important for the classification problem of the curves with one place at infinity.

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M. FUJIMOTO AND M. SUZUKI

Theorem 1 ([18]).
E0(i) The dual graph (Ef ) of Ef has the following form:
Ei1

Ei2

Eih

E1

Ej2

Ejh

ER

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(ii) f is non-constant only on ER and has the pole on Ef − ER .
(iii) The degree of f on ER is 1.
(iv) ER is the unique (−1)-curve in Ef .

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f is non-zero constant on T , T does not intersect Z and P . If T  = ∅, then T intersects only S. But since S( = ER ) is the last (−1)-curve on Mf , the relations of intersection among Z, P , S and T is one of the following two types:(I) P —S—Z
(II) P —S—T .
If Z  = ∅, then we get the contradiction as it is described in [18]. The similar argument
applies to the case of T  = ∅. Thus we get Z = ∅ and T = ∅. As a consequence, (Ef )has the above form.

ON

In (Ef ), let i1 , i2 , . . . , ih (resp. j0 , j1 , . . . , jh ) be the indices of the branch vertices (resp. the terminal vertices) from the left hand side, where j0 = 0 and j1 = 1. Let
MC be the surface obtained by the blowing-down of ER , ER−1 , . . . , Eih +1 from Mf .
For i (0 ≤ i ≤ ih ), we shall continue to denote by Ei the proper transform of Ei
in MC . Further, we set EC = E0 ∪ E1 ∪ · · · ∪ Eih . We shall call the pair (MC , EC )the compactification of C2 obtained by the minimal resolution of the singularity of Cat infinity. We set Lk = ik−1 <i≤ik Ei for each k (1 ≤ k ≤ h) like the following figure,where i0 = −1.

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Ej0

Ei1

Ei2

Eih
C

L1

L2

Ej1

Lh

Ej2

Ejh

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CONSTRUCTION OF AFFINE PLANE CURVES

5DEFINITION 1 (δ-sequence). Let δk (0 ≤ k ≤ h) be the order of the pole of fon Ejk . We shall call the sequence {δ0 , δ1 , . . . , δh } the δ-sequence of C (or of f ).
We have the following fact since degx f = m and degy f = n.
Fact 1. δ0 = n, δ1 = m.
DEFINITION 2 ((p, q)-sequence). Now, we assume that the weights of Lk is of
the following form:
−mr

−mr−1

−m1

Eik
Lk+1

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−n1

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Lk−1

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−ns−1
−ns
Ejk

We define the natural numbers pk , ak , qk , bk satisfying(pk , ak ) = 1, (qk , bk ) = 1, 0 < ak < pk , 0 < bk < qk ,pk
= m1 −
ak

1

m2 −

and

1

m3 − . .

.

−

1
mr

qk
= n1 −
bk

n2 −

1

.

1

n3 − . .

.

−

1
ns

N

We shall call the sequence {(p1 , q1 ), (p2 , q2 ), . . . , (ph , qh )} the (p, q)-sequence of C
(or of f ).

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We shall assume that f (x, y) is monic in y. We define approximate roots by
Abhyankar’s definition.

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DEFINITION 3 (approximate roots). Let f (x, y) be the defining polynomial, monicin y, of a curve with one place at infinity. Let {δ0 , δ1 , . . . , δh } be the δ-sequence of f .
We set n = degy f , dk = gcd{δ0 , δ1 , . . . , δk−1 } and nk = n/dk (1 ≤ k ≤ h + 1). Then, foreach k (1 ≤ k ≤ h + 1), a pair of polynomials (gk (x, y), ψk (x, y)) satisfying the following conditions is uniquely determined:(i) gk is monic in y and degy gk = nk ,(ii) degy ψk < n − nk ,(iii) f = gkdk + ψk .
We call this gk the k-th approximate root of f .

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M. FUJIMOTO AND M. SUZUKI

We can easily get the following fact from the definition of approximate roots.
Fact 2. We have
p/q

g1 = y +

ck x k ,gh+1 = fj =0
where ck ∈ C, p = degx f /d, q = degy f /d, d = gcd degx f, degy f and p/q isthe maximal integer l such that l ≤ p/q.

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The sequence of polynomials g0 := x, g1 , . . . , gh+1 is

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DEFINITION 4 (g-sequence).
called the g-sequence of f .

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Here, we denote by Ck the curve defined by gk (x, y) = 0 in C2 . The following
theorem about Ck plays a vital role in the main theorem.
Theorem 2. For each k (0 ≤ k ≤ h), Ck is also with one place at infinity. Further, its closure C k in MC intersects transversely Ejk , and does not intersect other
irreducible components of EC .
Suzuki [18] gave the algebrico-geometric proof of this theorem. We get the following theorem as a corollary of the above theorem.
Theorem 3. For each k (0 ≤ k ≤ h), gk has the pole of order δk on Eih .
The following lemma about approximate roots will be used in Theorem 6.

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Lemma 1. Let f be the defining polynomial, monic in y, of a curve with one
place at infinity. Let {δ0 , δ1 , . . . , δh } be the δ-sequence of f , and g0 , g1 , . . . , gh , gh+1
be the g-sequence of f . Then, gk (0 ≤ k ≤ h − 1) is also the k-th approximate root of
gj for any j with k < j < h + 1.
Proof. For example, see Proposition 2.2 in [5].

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3.

Intersection matrix and successive blow-up

Let M be a non-singular projective algebraic surface over complex number field,
and E be an algebraic curve on M. We shall assume that E1 , E2 , . . . , Es are irreducible components of E, and denote by IE the intersection matrix ((Ei · Ej ))i,j =1,...,sof E. The following lemma about the intersection matrix is well-known by Mumford.
Lemma 2. E is an exceptional set if and only if IE is negative definite.

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CONSTRUCTION OF AFFINE PLANE CURVES

7Let E1 be the (−1)-curve appeared by blowing-up at a point P0 on a surface M,
be the (−1)-curve appeared byand let P1 be a point on E1 . For i (≥ 1), let Ei+1

blowing-up at a point Pi , and let Pi+1 be the point on Ei+1
. We get {Pi }i=0,...,r and
{Ei }i=1,...,r by the above finite operations. In this paper we call this finite sequence of
blowing-ups a successive blow-up from P0 . Let M  be the surface obtained by a successive blow-up from P0 . For i (1 ≤ i ≤ r), we shall continue to denote by Ei therproper transform of Ei in M  . Further, we set E  = i=1 Ei and E  = det(−IE  ). We
have the following fact since E  is invariant under the successive blow-up.

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Fact 3. E  = 1.

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The following lemma is Lemma 1 in [18]. Here, we describe it because it is used
many times in the next section.

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Lemma 3. Let E1 , E2 , . . . , Er , Er+1 be the irreducible components of E and assume that the dual graph (E) is of the following linear type:
E1

E2

Er

Er+1

−n1

−n2

−nr

(ni ≥ 2).

Assume further that there exists a holomorphic function f on a neighborhood U ofri=1 Ei such that the zero divisor (f ) of f on U is written in the following form:r

mi Ei + mr+1 Er+1 ∩ U .

i=1

Let (pi , pi+1 ) be the coprime integers defined by the following continued fraction:

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pi+1
= ni −piD

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1

ni−1 − . .

(1 ≤ i ≤ r).

.

−

1
n1

Then, mi = m1 pi (1 ≤ i ≤ r + 1).
Now, consider a pair of natural numbers (p, q) with gcd(p, q) = 1, p > q > 0.
We can easily show that there exists a unique pair of natural numbers (a, b) with pq −aq − bp = 1, 0 < a < p, 0 < b < q.

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M. FUJIMOTO AND M. SUZUKI

We consider the following continued fractions for the above mentioned p, q, a, b:p= m1 −
a

1

,

1

m2 −

m3 − . .

.

−q= n1 −b1n2 −n3 − . .

1
mr

,

1
.

−

1
ns

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where mi ≥ 2 and nj ≥ 2.
Let (x, y) be the local coordinate for the neighborhood of a point P on M which
has P as the origin. Then,y-axis

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COLemma 4. we can construct a exceptional curve with the following weights bya successive blow-up from P .
−mr−1

−mr

−1

−m1

−n1

−ns−1

x-axis−ns

Proof. We consider the curve C defined by x p + y q = 0. The resolution graph at
origin of C is as follows:Ev
y-axis
−mv


Ev−1

E1

ET

E1

Eu−1

Eu

−mv−1

−m1

−1

−n1

−nu−1−nux-axisLet IE be the intersection matrix of the exceptional curve E corresponding to
the above dual graph. Here, we set

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p
= m1 −a1m2 − .
.

,

.

−

1
mu

q
= n1 −b1n2 − .
.

.

.

1
− 
nv

We get det(−IE ) = p  q  −a  q  −b p  . On the other hand, E is the exceptional curve obtained by a successive blow-up from origin. Therefore, we get det(−IE ) = 1 by Fact 3.
Thus p  q  − a  q  − b p  = 1.
As the above dual graph, let Ei (1 ≤ i ≤ u), ET , Ej (1 ≤ j ≤ v) be the irreducible
components of E. We denote by µi (1 ≤ i ≤ u) the zero order of the function x on Ei
and by µT the zero order of the function x on ET . Also, we denote by νj (1 ≤ j ≤ v)
the zero order of the function y on Ej and by νT the zero order of the function y
on ET . Since q = µT and µu = 1, we get q  = µT /µu = q by Lemma 3. As the sameway, we get p = p  . Thus pq − a  q − b p = 1. Further, it must be a = a  , b = b , since
0 < a  < p and 0 < b < q. Therefore, we get v = r, mi = mi (1 ≤ i ≤ r), u = s,Dtotal :

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CONSTRUCTION OF AFFINE PLANE CURVES

9nj = nj (1 ≤ j ≤ s) by the uniqueness of the expansion into continued fraction. As
a result, the assertion was proved.
4. Construction of a curve with one place at infinity
We set N = {n ∈ Z | n ≥ 0} and C∗ = C \ {0}. The following theorem about
δ-sequence and (p, q)-sequence is called Abhyankar-Moh’s Semigroup Theorem.

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Theorem 4 (Abhyankar-Moh). Let C be a non-linearlizable affine plane curvewith one place at infinity. Let {δ0 , δ1 , . . . , δh } be the δ-sequence of C and{(p1 , q1 ), . . . , (ph , qh )} be the (p, q)-sequence of C. We set dk = gcd{δ0 , δ1 , . . . , δk−1 }(1 ≤ k ≤ h + 1). We have then,(i) qk = dk /dk+1 , dh+1 = 1 (1 ≤ k ≤ h) ,(k = 1)
δ1,(ii) dk+1 pk =qk−1 δk−1 − δk (2 ≤ k ≤ h)(iii) qk δk ∈ Nδ0 + Nδ1 + · · · + Nδk−1 (1 ≤ k ≤ h) .

The following theorem gives the converse of the above theorem.

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Theorem 5 (Sathaye-Stenerson [16]). Let {δ0 , δ1 , . . . , δh } (h ≥ 1) be the sequence of h + 1 natural numbers. We set dk = gcd{δ0 , δ1 , . . . , δk−1 } (1 ≤ k ≤ h + 1)
and qk = dk /dk+1 (1 ≤ k ≤ h). Furthermore, suppose that the following conditions are
satisfied:(1) δ0 < δ1 ,(2) qk ≥ 2 (1 ≤ k ≤ h),(3) dh+1 = 1,(4) δk < qk−1 δk−1 (2 ≤ k ≤ h),(5) qk δk ∈ Nδ0 + Nδ1 + · · · + Nδk−1 (1 ≤ k ≤ h).
Then, there exists a curve with one place at infinity of the δ-sequence {δ0 , δ1 , . . . ,δh }.
Suzuki [18] gave an algebrico-geometric proof of the above two theorem by
the consideration of the resolution graph at infinity.

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DEFINITION 5 (Abhyankar-Moh’s condition). We shall call the conditions (1)–(5)concerning {δ0 , δ1 , . . . , δh } in Theorem 5 Abhyankar-Moh’s condition.
Theorem 6. Let {δ0 , δ1 , . . . , δh } (h ≥ 1) be the sequence of h + 1 natural numbers satisfying Abhyankar-Moh’s condition. Set dk = gcd{δ0 , δ1 , . . . , δk−1 } (1 ≤ k ≤h + 1) and qk = dk /dk+1 (1 ≤ k ≤ h). Then,(i) the defining polynomial f , monic in y, of a curve with one place at infinity of2002/11/5(20:33)

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M. FUJIMOTO AND M. SUZUKI

the δ-sequence {δ0 , δ1 , . . . , δh } has the following form using the approximate roots
g0 , g1 , . . . , gh of f :ᾱ

h−1
f = ghqh + aᾱ0 ᾱ1 ···ᾱh−1 g0ᾱ0 g1ᾱ1 · · · gh−1+cα0 α1 ···αh g0α0 g1α1 · · · ghαh

(α0 , α1 , ..., αh )∈

where aᾱ0 ᾱ1 ···ᾱh−1 ∈ C∗ , cα0 α1 ···αh ∈ C, (ᾱ0 , ᾱ1 , . . . , ᾱh−1 ) is the sequence of h nonnegative integers satisfyingh−1ᾱi δi = qh δh , ᾱi < qi (0 < i < h)

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i=0

and
h+1 

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=



h


αi δi < qh δh .
 αi < qi (0 < i < h), αh < qh − 1,


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

i=0

(ii) Conversely, let gh be the defining polynomial, monic in y, of a curve with one
place at infinity of the δ-sequence {δ0 /qh , δ1 /qh , . . . , δh−1 /qh }, and g0 , g1 , . . . , gh−1
be the approximate roots of gh . For any non-zero complex number aᾱ0 ᾱ1 ···ᾱh−1 corresponding to the sequence of h non-negative integers (ᾱ0 , ᾱ1 , . . . , ᾱh−1 ) satisfyingh−1ᾱi δi = qh δh ,ᾱi < qi (0 < i < h)i=0

and any complex numbers cα0 α1 ···αh corresponding to the sequences of h + 1 nonnegative integers (α0 , α1 , . . . , αh ) satisfying
αi δi < qh δh ,αi < qi (0 < i < h),

αh < qh − 1,

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h

i=0

O

we considerᾱDtotal :h−1
f = ghqh + aᾱ0 ᾱ1 ···ᾱh−1 g0ᾱ0 g1ᾱ1 · · · gh−1+cα0 α1 ···αh g0α0 g1α1 · · · ghαh

(α0 , α1 , ..., αh )∈

where

=



h


αi δi < qh δh .
 αi < qi (0 < i < h), αh < qh − 1,


h+1 

(α0 , α1 , . . . , αh ) ∈ N

i=0

Then, the curve defined by f = 0 is a curve with one place at infinity of theδ-sequence {δ0 , δ1 , . . . , δh }, and has the approximate roots g0 , g1 , . . . , gh .

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CONSTRUCTION OF AFFINE PLANE CURVES

11

Proof of Theorem 6. We shall prove (i). By the procedure described in the proofof Proposition 10 in [18], using the approximate roots g0 , g1 , . . . , gh of f and the sethof h + 1 non-negative integers (α0 , α1 , . . . , αh ) with max{ i=0 αi δi } = qh δh , we canwrite f as follows:f =cα0 α1 ···αh g0α0 g1α1 · · · ghαh + ghqh ,cα0 α1 ···αh ∈ C.

αi <qi (1≤i≤h)Set δ0 = q and δ1 = p. We can write f as follows:

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CASE h = 1.

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Here, we suppose f = ghqh + ghqh −1 . We have degy ghqh −1 = nh (qh − 1) = degy f − nh =n − nh . But this is a contradiction, since gh is h-th approximate root of f . Thus we
get αh < qh − 1. By Theorem 4(iii) and the uniqueness of {αi }i=0,...,h (e.g., Lemma 7
h−1
in [18]), we have {ᾱi }i=0,...,h−1 with i=0 ᾱi δi = qh δh . As a result, (i) was proved.
We shall prove (ii).



f = y q + ax p +cαβ x α y β ,a ∈ C∗ ,cαβ ∈ C.

qα+pβ<pq

The curve defined by f = 0 has one place at infinity of the δ-sequence {q, p} by theconsideration of Newton boundary.

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CASE h ≥ 2. Set δi /qh = δ̃i (0 ≤ i ≤ h − 1). We denote by Ck the curve definedby gk = 0 for each k with 0 ≤ k ≤ h. Further, we shall denote by (M̃, Ẽ) the compactification of C2 obtained by the minimal resolution of Ch at infinity. Let C̃k be theproper transform of Ck on M̃ and Ẽi be the irreducible components of Ẽ. (The wayof numbering about indices is same as Section 2.) By Theorem 2, C̃k has one place at
infinity and intersects transversely Ẽjk (0 ≤ k ≤ h − 1).
Let Q be the intersection point of C̃h and Ẽih−1 . Set ph = qh−1 δh−1 − δh . (ph > 0
since Abhyankar-Moh’s condition (4).) We have gcd(ph , qh ) = 1 from gcd(qh , δh ) =dh+1 = 1 and get a unique pair of natural numbers (ah , bh ) with ph qh − ah qh − bh ph =1, 0 < ah < ph , 0 < bh < qh . We define {mi }i=1,...,r , {nj }j =1,...,s using the following
expansion into continued fractions by ph , ah , qh , bh :Dtotal :

ph= m1 −
ah

1
m2 −

,

1
m3 − . .

.

−

1
mr

qh
= n1 −
bh

1
n2 −

.

1
n3 − . .

.

−

1
ns

By Lemma 4, we can obtain the following branch Lh such that Ch intersects
transversely Ejh using the successive blow-up from Q:

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M. FUJIMOTO AND M. SUZUKI

Eih

Ejh
.

Lh−1
−mr

−mr−1

−m1

−1

−n1

−ns−1

−ns

v
uqh−1 δ̃h−1

× (non-const).

CO

gh =

PY

Let M be the surface thus obtained, E be the total transform of Ẽ on M. We denoteby Ei (resp. C k ) the proper transform of Ẽi (resp. C̃k ).
Set dk = gcd{δ0 , δ1 , . . . , δk−1 } (1 ≤ k ≤ h + 1) and qk = dk /dk+1 (1 ≤ k ≤ h).
By Theorem 3, gk has the pole of order δ̃k on Eih−1 for each k (0 ≤ k ≤ h − 1). Thus
gk has the pole of order δ̃k on Ejh and of order qh δ̃k (= δk ) on Eih . On the other hand,
gh has the pole of order δh on Eih . In fact, we can write gh on a neighborhood of Q
as follows:

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Now, we consider the curve C defined by f = 0. Set φ = f − ghqh and  = φ/ghqh .
Since the both of ghqh and φ has the pole of order qh δh on Eih ,  is non-constant or
constant( = 0) on Eih .
Let A (resp. B) be the closure of the connected component of E − Eih which contains E0 (resp. Ejh ). Let Pgh be the pole divisor of gh on M, and D be its restriction
to A. Here, let F1 be the irreducible component of A intersecting Eih . Since gh has
the pole of order δh on Eih , we have (D · F1 ) < 0. Also, since (D · Ei ) = 0 for any
Ei with Ei  = F1 , using Proposition 2 in [6], the intersection matrix of A is negative definite. Thus it follows that A is exceptional set.  is holomorphic on A since
A ∩ C h = ∅. On the other hand,degy g0α0 g1α1 · · · ghαhhi=0αi ni =

N

=

h


αi ni

i=1

O

= α1 + α2 q1 + α3 q2 q1 + · · · + αh qh−1 · · · q1

< (q1 − 1) + (q2 − 1)q1 + (q3 − 1)q2 q1 + · · · + (qh − 1)qh−1 · · · q1

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total :

= qh qh−1 · · · q1 − 1< qh qh−1 · · · q1 = qh nh = degy ghqh .
Therefore, we get degy φ < degy ghqh . Hence,  = 0 on E0 . Further,  = 0 on A, since
A is compact. As a result, it must be that  is non-constant on Eih .
Let P be the pole divisor of  on M. We denote by B1 , B2 , . . . , Bs the irreducible components of B in order from the component intersecting Eih . Since 
has the pole on B1 and C h , the support of P is B ∪ C h and we can write P =

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CONSTRUCTION OF AFFINE PLANE CURVES

qh C h +13si=1 µi Bi (µi > 0). By

1

ns −

ns−1 − .

=
..

−

qh
b

1
n1

1 (i = ih ).
0 (i  = ih )

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and Lemma 3, we get µ1 qh = qh , where µ1 is the pole order of  on B1 . Hence,µ1 = 1. This implies that  is a rational function of degree 1 on Eih . Therefore,
the curve defined by  = −1 intersects transversely Eih at only one point. Since thecurve  = −1 coincides with C, we get

As a result, C has one place at infinity.
We have f = ghqh on A, since  = 0 on A. Hence, f has the pole of the sameorder as ghqh on each irreducible component of A. In particular, f has the pole of order qh δ̃k = δk on each Eik (0 ≤ k ≤ h − 1). Since  is non-constant on Eih , f
has the pole of the same order as ghqh on Eih . Since the value of its pole order isqh δh , using Lemma 3, it follows that f has the pole of order δh on Ejh . Consequently,{δ0 , δ1 , . . . , δh } is the δ-sequence of f .
Finally, we show that g0 , g1 , . . . , gh are the approximate roots of f . By
degy g0α0 g1α1 · · · ghαh

= n0 α0 + n1 α1 + · · · + nh αh≤ n1 (q1 − 1) + n2 (q2 − 1) + · · · + nh−1 (qh−1 − 1) + nh (qh − 2)
= −n1 + nh qh − nh < n − nh ,ON

gh is h-th approximate root of f . Therefore, by Lemma 1, g0 , g1 , . . . , gh are the approximate roots of f .

Dtotal :

The following theorem is the main theorem in this paper, and is obtained by using
Theorem 6 inductively.
Theorem 7. Let {δ0 , δ1 , . . . , δh } (h ≥ 1) be a sequence of natural numbers satisfying Abhyankar-Moh’s condition (see Definition 5). Set dk = gcd{δ0 , δ1 , . . . , δk−1 }(1 ≤ k ≤ h + 1) and qk = dk /dk+1 (1 ≤ k ≤ h).

2002/11/5(20:33)

14

M. FUJIMOTO AND M. SUZUKI(1) We define gk (0 ≤ k ≤ h + 1) as follows:
g0 = x,




p/q



δ1
δ0


=
y
+
cj x j , cj ∈ C, p = , q = ,
g

1


d
d
2
2

j =0
ᾱi−1
qi
ᾱ0 ᾱ1
gi+1 = gi + aᾱ0 ᾱ1 ···ᾱi−1 g0 g1 · · · gi−1







+
cα0 α1 ···αi g0α0 g1α1 · · · giαi ,(α0 , α1 , ..., αi )∈iaᾱ0 ᾱ1 ···ᾱi−1 ∈ C∗ , cα0 α1 ···αi ∈ C (1 ≤ i ≤ h),ᾱj δj = qi δi , ᾱj < qj (0 < j < i)

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j =0

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i−1


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where (ᾱ0 , ᾱ1 , . . . , ᾱi−1 ) is the sequence of i non-negative integers satisfying

and






i



αj δj < qi δi .
i = (α0 , α1 , . . . , αi ) ∈ Ni+1  αj < qj (0 < j < i), αi < qi − 1,



j =0
Then, g0 , g1 , . . . , gh are approximate roots of f (= gh+1 ), and f is the defining polynomial, monic in y, of a curve with one place at infinity of the δ-sequence{δ0 , δ1 , . . . , δh }.
(2) The defining polynomial f , monic in y, of a curve with one place at infinity ofthe δ-sequence {δ0 , δ1 , . . . , δh } is obtained by the procedure of (1), and the values ofparameters {aᾱ0 ᾱ1 ···ᾱi−1 }1≤i≤h and {cα0 α1 ···αi }0≤i≤h are uniquely determined for f .

N

The above theorem gives normal forms of defining polynomials of curves with
one place at infinity and the method of construction of their defining polynomials.

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Corollary 1. Let {δ0 , δ1 , . . . , δh } (h ≥ 1) be a sequence of natural numbers satisfying Abhyankar-Moh’s condition. The moduli space of the curve C with one placeat infinity of the δ-sequence {δ0 , δ1 , . . . , δh } is isomorphic toDtotal :(C∗ )h × Cb ,where b is the total number of parameters {cα0 α1 ···αi }0≤i≤h appeared in the defining
polynomial, monic in y, of C obtained in Theorem 7.
Proof. We consider the defining polynomial f , monic in y, of the curve C with
one place at infinity of the δ-sequence {δ0 , δ1 , . . . , δh }. We denote by a the number

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CONSTRUCTION OF AFFINE PLANE CURVES

15

of non-zero parameters in f and by b the number of others. By Theorem 7, the moduli space of C is (C∗ )a × Cb . f has h + 2 polynomials g0 , g1 , . . . , gh+1 . Here, bothof g0 and g1 do not have non-zero parameter. Also, gi+1 (1 ≤ i ≤ h) has exactly one
non-zero parameter because the sequence of i +1 non-negative integers (α0 , α1 , . . . , αi )iwith j =0 αj δ̃j = qi δ̃i is determined uniquely. As a result, we get a = h.
By the above results, we can easily get an algorithm generating the defining polynomial and computing the moduli space from a δ-sequence. We will introduce them in
the next section.

PY

5. Algorithms

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Using Theorem 7, the following algorithm generating the defining polynomial of
the curve with one place at infinity from a δ-sequence is obtained.

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Algorithm 1: generating polynomial
Input: δ-sequence {δ0 , δ1 , . . . , δh }Output: the defining polynomial f (x, y) of the curve with one place at infinity of theδ-sequence {δ0 , δ1 , . . . , δh }OND ← [δh , δh−1 , . . . , δ0 ]dk ← gcd{δ0 , δ1 , . . . , δk−1 } (1 ≤ k ≤ h + 1)
Q ← [qh , . . . , q1 ] where qk = dk /dk+1 (1 ≤ k ≤ h)DL ← cons(D, [ ])QL ← cons(Q, [ ])m←h+1
while m  = 2 do
T ← reverse(cdr(D))
D←[]
while T  = [ ] doD ← cons(car(T )/car(Q), D)T ← cdr(T )endDL ← cons(D, DL)Q ← cdr(Q)QL ← cons(Q, QL)m ← length(D)endAL ← [x]D ← car(DL)l ← car(D)/car(cdr(D))
l
g1 ← y + j =0 cj x jDtotal :

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M. FUJIMOTO AND M. SUZUKI

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AL ← cons(g1 , AL)
while DL  = [ ] doD ← car(DL)Q ← car(QL)
q0 ← car(Q) × car(D)/car(reverse(D)) + 1
L ← append(Q, [q0 ])k ← length(D) − 1, i.e., D = [δ̄k , . . . , δ̄0 ], L = [qk , . . . , q0 ].
(ᾱ0 , ᾱ1 , . . . , ᾱk−1 ) ← the sequence of non-negative integers withk−1
i=0 ᾱi δ̄i = δ̄k qk , ᾱi < qi (0 ≤ i ≤ k − 1), δ̄i ∈ D and qi ∈ L{(α0 , α1 , . . . , αk )} ← the set of sequences of non-negative integers withk< qi (0 ≤ i < k), αk < qk − 1, δ̄i ∈ D and qi ∈ L
i=0 αi δ̄i < δ̄k qk , αi 
k
k−1
qk
gk+1 ← gk + aᾱ0 ,ᾱ1 ,...,ᾱk−1 i=0 giᾱi + cα0 ,α1 ,...,αk i=0 giαi
AL ← cons(gk+1 , AL)
DL ← cdr(DL)QL ← cdr(QL)
end
return car(AL)ONS UPPLEMENTATION :• [ . . . ] := A list. (This is a data structure with ordered elements.)• p := The maximal integer n such that n ≤ p.
• car(L) := The first element of a given non-null list L.
• cdr(L) := The list obtained by removing the first element of a given non-null listL.
• cons(A, L) := The list obtained by adding an element A to the top of a given listL.
• reverse(L) := The reversed list of a given list L.
• append(L1 , L2 ) := The list obtained by adding all elements in a list L2 according
to the order as it is to the last element in a list L1 .
• length(L) := The number of elements of a given list L.
• a∗,∗,...,∗ is a parameter in C∗ .
• c∗,∗,...,∗ is a parameter in C.
The moduli space of f is obtained by counting the numbers of {a∗,∗,...,∗ } and{c∗,∗,...,∗ } in f which the above algorithm outputted. But we can compute the moduli space from a δ-sequence without generating the defining polynomial. The following
algorithm directly compute the moduli space from a δ-sequence.

D

total :

Algorithm 2: computation of moduli space
Input: δ-sequence {δ0 , δ1 , . . . , δh }Output: [M, N ] ( This means the moduli space (C∗ )M × CN . )D ← [δh , δh−1 , . . . , δ0 ]

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17

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dk ← gcd{δ0 , δ1 , . . . , δk−1 } (1 ≤ k ≤ h + 1)
Q ← [qh , . . . , q1 ] where qk = dk /dk+1 (1 ≤ k ≤ h)Q ← cons(1, Q)M←hN ←0while true dok ← length(D) − 1, i.e., D = [δ̄k , . . . , δ̄0 ]D ← [δ̄k /car(Q), δ̄k−1 /car(Q), . . . , δ̄0 /car(Q)]Q ← cdr(Q)q0 ← car(Q) × car(D)/car(reverse(D)) + 1
L ← append(Q, [q0 ]), i.e., L = [qk , . . . , q0 ]kn ← the number of (α0 , α1 , . . . , αk ) with i=0 αi δ̄i < car(Q) × car(D),αi < qi (0 ≤ i ≤ k − 1), αk < qk − 1, δ̄i ∈ D and qi ∈ L
N ←N +n
if length(D) = 2 then breakD ← cdr(D)endN ← N + p/q + 1
return [M, N ]6. Polynomial curve

6.1. Abhyankar’s question.
question.

In this section, we will introduce Abhyankar’s

DEFINITION 6 (planar semigroup). Let {δ0 , δ1 , . . . , δh } (h ≥ 1) be a sequenceof natural numbers satisfying Abhyankar-Moh’s condition. A semigroup generated by{δ0 , δ1 , . . . , δh } is said to be a planar semigroup.

O

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DEFINITION 7 (polynomial curve). Let C be an algebraic curve defined byf (x, y) = 0, where f (x, y) is an irreducible polynomial in C[x, y]. We call C a polynomial curve, if C has a parametrisation x = x(t), y = y(t), where x(t) and y(t) arepolynomials in C[t].

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Abhyankar’s Question. Let  be a planar semigroup. Is there a polynomial
curve with δ-sequence generating ?
This question is still open. Moh [10] showed that there is no polynomial
curve with δ-sequence {6, 8, 3}. But there is a polynomial curve (x, y) = (t 3 , t 8 )with δ-sequence {3, 8} which generates the same semigroup as above. SathayeStenerson [16] proved that the semigroup generated by {6, 22, 17} has no otherδ-sequence generating the same semigroup, and proposed the following conjecture for2002/11/5(20:33)

18

M. FUJIMOTO AND M. SUZUKI

this question.
Sathaye-Stenerson’s Conjecture. There is no polynomial curve having theδ-sequence {6, 22, 17}.
By Algorithm 1, the defining polynomial of the curve with one place at infinityof the δ-sequence {6, 22, 17} as follows:f = (g22 + a2,1 x 2 g1 ) + c5,0,0 x 5 + c4,0,0 x 4 + c3,0,0 x 3 + c2,0,0 x 2

PY+c1,1,0 xg1 + c1,0,0 x + c0,1,0 g1 + c0,0,0

CO

where
g1 = y + c3 x 3 + c2 x 2 + c1 x + c0 ,

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g2 = (g13 + a11 x 11 ) + c10,0 x 10 + c9,0 x 9 + c8,0 x 8 + (c7,1 g1 + c7,0 )x 7+(c6,1 g1 + c6,0 )x 6 + (c5,1 g1 + c5,0 )x 5 + (c4,1 g1 + c4,0 )x 4+(c3,1 g1 + c3,0 )x 3 + (c2,1 g1 + c2,0 )x 2 + (c1,1 g1 + c1,0 )x + c0,1 g1 + c0,0 .
This result gives us a new approach to investigate the curve with one place at infinity of the δ-sequence {6, 22, 17} using a computer algebra system.

N

6.2. Computation of moduli space. Suzuki gave an algorithm generating
the list of δ-sequences of curves with one place at infinity, and implemented on
a computer. From the list of δ-sequences obtained by Suzuki, we could get normal
forms and moduli spaces of curves with one place at infinity of genus ≤ 100 by using
the algorithm introduced in previous section. As a result, we could verify the resultof Nakazawa-Oka [11].
The following is the list of moduli spaces of curves with one place at infinity forthe cases genus ≤ 30.

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EXAMPLE 1.

The case[7,[4,6,11],[2,15]]
means that the moduli space of the curve with one place at infinity of genus 7 andthe δ-sequence {4, 6, 11} is isomorphic to (C∗ )2 × C15 .

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[1,[2,3],[1,5]],[2,[2,5],[1,8]],[3,[2,7],[1,11]],[3,[3,4],[1,9]],[3,[4,6,3],[2,7]],[4,[2,9],[1,14]],[4,[3,5],[1,11]],[4,[4,6,5],[2,9]],[4,[6,9,2],[2,7]],[5,[2,11],[1,17]],[5,[4,6,7],[2,11]],[6,[2,13],[1,20]],[6,[3,7],[1,15]],[6,[4,5],[1,14]],[6,[4,6,9],[2,13]],[6,[6,9,4],[2,10]],[6,[4,10,5],[2,11]],[7,[2,15],[1,23]],[7,[3,8],[1,17]],[7,[4,6,11],[2,15]],[7,[6,9,5],[2,12]],[7,[8,12,3],[2,10]],[7,[10,15,2],[2,9]],[7,[4,10,7],[2,13]],[7,[6,15,2],[2,10]],[7,[6,8,3],[2,10]],[7,[8,12,6,3],[3,8]],

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CONSTRUCTION OF AFFINE PLANE CURVES

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19

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[16,[8,20,10,9],[3,15]],[17,[2,35],[1,53]],[17,[10,15,7],[2,22]],[17,[8,20,7],[2,19]],[17,[14,35,2],[2,14]],[17,[4,14,23],[2,31]],[17,[10,35,2],[2,15]],[17,[4,18,19],[2,29]],[17,[4,22,15],[2,27]],[17,[6,8,23],[2,29]],[17,[6,10,19],[2,26]],[17,[8,12,10,19],[3,24]],[17,[8,12,14,15],[3,21]],[17,[8,20,14,7],[3,15]],[18,[2,37],[1,56]],[18,[3,19],[1,39]],[18,[4,13],[1,34]],[18,[6,9,16],[2,30]],[18,[6,15,13],[2,26]],[18,[4,14,25],[2,33]],[18,[6,21,10],[2,22]],[18,[4,18,21],[2,31]],[18,[4,22,17],[2,29]],[18,[6,33,4],[2,21]],[18,[4,26,13],[2,27]],[18,[9,12,10],[2,20]],[18,[6,10,21],[2,28]],[18,[6,14,13],[2,22]],[18,[6,16,9],[2,21]],[18,[8,10,13],[2,21]],[18,[12,18,9,10],[3,18]],[18,[8,12,14,17],[3,23]],[18,[12,18,21,4],[3,13]],[18,[8,12,18,13],[3,20]],[18,[12,18,10,9],[3,15]],[18,[12,18,8,13],[3,17]],[18,[16,24,6,9],[3,14]],[18,[8,20,10,13],[3,18]],[18,[12,16,6,9],[3,14]],[18,[16,24,12,6,9],[4,12]],[19,[2,39],[1,59]],[19,[3,20],[1,41]],[19,[6,9,17],[2,32]],[19,[8,12,11],[2,28]],[19,[10,15,8],[2,25]],[19,[14,21,5],[2,21]],[19,[20,30,3],[2,18]],[19,[26,39,2],[2,17]],[19,[6,15,14],[2,28]],[19,[4,14,27],[2,35]],[19,[6,21,11],[2,24]],[19,[4,18,23],[2,33]],[19,[6,27,8],[2,21]],[19,[4,22,19],[2,31]],[19,[4,26,15],[2,29]],[19,[6,39,2],[2,22]],[19,[9,12,11],[2,22]],[19,[15,20,3],[2,13]],[19,[6,10,23],[2,30]],[19,[9,15,8],[2,18]],[19,[12,20,3],[2,14]],[19,[6,14,15],[2,24]],[19,[6,16,11],[2,22]],[19,[6,20,3],[2,22]],[19,[8,10,15],[2,23]],[19,[12,18,9,11],[3,20]],[19,[20,30,15,3],[3,11]],CO[13,[18,27,6,2],[3,9]],[14,[2,29],[1,44]],[14,[5,8],[1,26]],[14,[8,20,5],[2,15]],[14,[4,14,17],[2,25]],[14,[4,18,13],[2,23]],[14,[6,8,17],[2,23]],[14,[6,10,13],[2,20]],[14,[8,10,5],[2,16]],[14,[8,12,10,13],[3,18]],[14,[8,20,10,5],[3,13]],[15,[2,31],[1,47]],[15,[3,16],[1,33]],[15,[4,11],[1,29]],[15,[6,7],[1,27]],[15,[6,9,13],[2,25]],[15,[10,15,6],[2,19]],[15,[16,24,3],[2,15]],[15,[6,15,10],[2,21]],[15,[4,14,19],[2,27]],[15,[6,21,7],[2,18]],[15,[4,18,15],[2,25]],[15,[6,27,4],[2,18]],[15,[4,22,11],[2,23]],[15,[6,8,19],[2,25]],[15,[9,12,7],[2,16]],[15,[12,16,3],[2,12]],[15,[6,10,15],[2,22]],[15,[6,14,7],[2,18]],[15,[6,16,3],[2,18]],[15,[12,18,9,7],[3,14]],[15,[16,24,12,3],[3,10]],[15,[8,12,10,15],[3,20]],[15,[12,18,15,4],[3,11]],[15,[8,12,14,11],[3,17]],[15,[18,27,6,4],[3,11]],[15,[16,24,6,3],[3,11]],[15,[12,16,6,3],[3,11]],[15,[16,24,12,6,3],[4,9]],[16,[2,33],[1,50]],[16,[3,17],[1,35]],[16,[5,9],[1,29]],[16,[6,9,14],[2,27]],[16,[8,12,9],[2,23]],[16,[12,18,5],[2,19]],[16,[14,21,4],[2,17]],[16,[22,33,2],[2,15]],[16,[6,15,11],[2,23]],[16,[10,25,4],[2,15]],[16,[4,14,21],[2,29]],[16,[6,21,8],[2,19]],[16,[4,18,17],[2,27]],[16,[4,22,13],[2,25]],[16,[6,33,2],[2,19]],[16,[6,8,21],[2,27]],[16,[9,12,8],[2,17]],[16,[6,10,17],[2,24]],[16,[9,15,5],[2,15]],[16,[6,14,9],[2,19]],[16,[8,10,9],[2,18]],[16,[12,18,9,8],[3,15]],[16,[8,12,10,17],[3,22]],[16,[12,18,15,5],[3,12]],[16,[8,12,14,13],[3,19]],[16,[8,12,18,9],[3,17]],[16,[12,18,10,5],[3,13]],[16,[12,18,8,9],[3,14]],

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AT oo
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IS[8,[2,17],[1,26]],[8,[4,10,9],[2,15]],[9,[2,19],[1,29]],[9,[3,10],[1,21]],[9,[4,7],[1,19]],[9,[6,9,7],[2,15]],[9,[10,15,3],[2,11]],[9,[4,10,11],[2,17]],[9,[6,15,4],[2,12]],[9,[4,14,7],[2,15]],[9,[6,8,7],[2,13]],[9,[6,10,3],[2,12]],[9,[8,12,6,7],[3,11]],[10,[2,21],[1,32]],[10,[3,11],[1,23]],[10,[5,6],[1,20]],[10,[6,9,8],[2,17]],[10,[8,12,5],[2,14]],[10,[14,21,2],[2,11]],[10,[4,10,13],[2,19]],[10,[6,15,5],[2,13]],[10,[4,14,9],[2,17]],[10,[6,21,2],[2,13]],[10,[6,8,9],[2,15]],[10,[6,10,5],[2,13]],[10,[8,12,6,9],[3,13]],[10,[8,12,10,5],[3,11]],[11,[2,23],[1,35]],[11,[10,15,4],[2,14]],[11,[4,10,15],[2,21]],[11,[4,14,11],[2,19]],[11,[6,8,11],[2,17]],[11,[8,12,6,11],[3,15]],[12,[2,25],[1,38]],[12,[3,13],[1,27]],[12,[4,9],[1,24]],[12,[5,7],[1,23]],[12,[6,9,10],[2,20]],[12,[4,10,17],[2,23]],[12,[6,15,7],[2,16]],[12,[10,25,2],[2,12]],[12,[4,14,13],[2,21]],[12,[6,21,4],[2,15]],[12,[4,18,9],[2,19]],[12,[6,8,13],[2,19]],[12,[9,12,4],[2,12]],[12,[6,10,9],[2,16]],[12,[12,18,9,4],[3,10]],[12,[8,12,10,9],[3,14]],[12,[12,18,4,9],[3,12]],[13,[2,27],[1,41]],[13,[3,14],[1,29]],[13,[6,9,11],[2,22]],[13,[8,12,7],[2,19]],[13,[14,21,3],[2,14]],[13,[18,27,2],[2,13]],[13,[4,10,19],[2,25]],[13,[6,15,8],[2,18]],[13,[4,14,15],[2,23]],[13,[4,18,11],[2,21]],[13,[6,27,2],[2,16]],[13,[6,8,15],[2,21]],[13,[6,10,11],[2,18]],[13,[6,14,3],[2,16]],[13,[8,12,10,11],[3,16]],[13,[8,12,14,7],[3,14]],[13,[12,18,4,11],[3,14]],D

total :

2002/11/5(20:33)

20

M. FUJIMOTO AND M. SUZUKI

O

N

OT co
py
D
U for
PL
IC pr
AT oo
E frea
TH d
IS

CO

PY

[19,[12,18,15,8],[3,16]],[21,[18,27,6,10],[3,17]],[22,[16,24,20,10,5],[4,12]],[19,[8,12,14,19],[3,25]],[21,[12,18,10,15],[3,20]],[23,[2,47],[1,71]],[19,[8,12,18,15],[3,22]],[21,[12,18,8,19],[3,23]],[23,[8,20,11],[2,28]],[19,[8,12,22,11],[3,20]],[21,[18,27,12,4],[3,12]],[23,[14,35,4],[2,18]],[19,[18,27,6,8],[3,14]],[21,[12,18,14,7],[3,17]],[23,[4,18,31],[2,41]],[19,[12,18,10,11],[3,17]],[21,[16,24,6,15],[3,18]],[23,[4,22,27],[2,39]],[19,[12,18,8,15],[3,19]],[21,[20,30,4,15],[3,17]],[23,[4,26,23],[2,37]],[19,[16,24,6,11],[3,15]],[21,[8,20,10,19],[3,24]],[23,[4,30,19],[2,35]],[19,[20,30,6,3],[3,12]],[21,[12,30,15,4],[3,13]],[23,[6,14,23],[2,32]],[19,[8,20,10,15],[3,20]],[21,[8,20,14,15],[3,21]],[23,[6,16,19],[2,29]],[19,[8,20,14,11],[3,18]],[21,[8,28,14,7],[3,17]],[23,[8,10,23],[2,31]],[19,[12,16,6,11],[3,15]],[21,[12,16,14,7],[3,15]],[23,[8,14,11],[2,23]],[19,[12,20,6,3],[3,13]],[21,[16,24,12,14,7],[4,13]], [23,[8,12,14,27],[3,33]],[19,[16,24,12,6,11],[4,13]], [22,[2,45],[1,68]],[23,[8,12,18,23],[3,30]],[20,[2,41],[1,62]],[22,[3,23],[1,47]],[23,[8,12,22,19],[3,27]],[20,[5,11],[1,35]],[22,[5,12],[1,38]],[23,[12,18,10,19],[3,24]],[20,[8,20,9],[2,23]],[22,[8,12,13],[2,32]],[23,[12,18,8,23],[3,27]],[20,[10,25,6],[2,19]],[22,[14,21,6],[2,25]],[23,[16,24,6,19],[3,22]],[20,[4,18,25],[2,35]],[22,[16,24,5],[2,23]],[23,[20,30,4,19],[3,21]],[20,[4,22,21],[2,33]],[22,[30,45,2],[2,19]],[23,[8,20,14,19],[3,25]],[20,[4,26,17],[2,31]],[22,[6,15,17],[2,33]],[23,[8,20,22,11],[3,21]],[20,[6,10,25],[2,32]],[22,[10,25,7],[2,22]],[23,[8,28,14,11],[3,19]],[20,[6,14,17],[2,26]],[22,[12,30,5],[2,19]],[24,[2,49],[1,74]],[20,[8,10,17],[2,25]],[22,[18,45,2],[2,16]],[24,[3,25],[1,51]],[20,[8,12,14,21],[3,27]],[22,[6,21,14],[2,29]],[24,[4,17],[1,44]],[20,[8,12,18,17],[3,24]],[22,[4,18,29],[2,39]],[24,[5,13],[1,41]],[20,[12,18,8,17],[3,21]],[22,[6,27,11],[2,25]],[24,[7,9],[1,39]],[20,[8,20,10,17],[3,22]],[22,[10,45,2],[2,18]],[24,[6,15,19],[2,36]],[20,[8,20,18,9],[3,18]],[22,[4,22,25],[2,37]],[24,[10,25,8],[2,24]],[21,[2,43],[1,65]],[22,[6,33,8],[2,24]],[24,[6,21,16],[2,32]],[21,[3,22],[1,45]],[22,[4,26,21],[2,35]],[24,[8,28,9],[2,24]],[21,[4,15],[1,39]],[22,[4,30,17],[2,33]],[24,[14,49,2],[2,17]],[21,[7,8],[1,35]],[22,[6,45,2],[2,25]],[24,[4,18,33],[2,43]],[21,[10,15,9],[2,28]],[22,[9,12,14],[2,27]],[24,[6,27,13],[2,28]],[21,[12,18,7],[2,25]],[22,[6,10,29],[2,36]],[24,[4,22,29],[2,41]],[21,[18,27,4],[2,21]],[22,[9,15,11],[2,22]],[24,[6,33,10],[2,26]],[21,[22,33,3],[2,19]],[22,[12,20,5],[2,16]],[24,[4,26,25],[2,39]],[21,[6,15,16],[2,31]],[22,[6,14,21],[2,30]],[24,[4,30,21],[2,37]],[21,[6,21,13],[2,27]],[22,[6,16,17],[2,27]],[24,[6,45,4],[2,27]],[21,[8,28,7],[2,20]],[22,[6,20,9],[2,24]],[24,[4,34,17],[2,35]],[21,[10,35,4],[2,18]],[22,[8,10,21],[2,29]],[24,[9,12,16],[2,30]],[21,[4,18,27],[2,37]],[22,[12,15,5],[2,17]],[24,[12,16,9],[2,22]],[21,[6,27,10],[2,24]],[22,[10,12,5],[2,21]],[24,[9,15,13],[2,25]],[21,[4,22,23],[2,35]],[22,[12,18,9,14],[3,25]],[24,[15,25,3],[2,15]],[21,[4,26,19],[2,33]],[22,[12,18,15,11],[3,20]],[24,[6,14,25],[2,34]],[21,[6,39,4],[2,24]],[22,[16,24,20,5],[3,13]],[24,[9,21,7],[2,20]],[21,[4,30,15],[2,31]],[22,[8,12,14,25],[3,31]],[24,[6,16,21],[2,31]],[21,[9,12,13],[2,25]],[22,[12,18,21,8],[3,17]],[24,[6,20,13],[2,27]],[21,[12,16,7],[2,18]],[22,[8,12,18,21],[3,28]],[24,[6,22,9],[2,26]],[21,[15,20,4],[2,15]],[22,[8,12,22,17],[3,25]],[24,[8,10,25],[2,33]],[21,[6,10,27],[2,34]],[22,[18,27,6,11],[3,18]],[24,[12,18,9,16],[3,28]],[21,[9,15,10],[2,21]],[22,[30,45,10,2],[3,11]],[24,[16,24,12,9],[3,20]],[21,[6,14,19],[2,28]],[22,[12,18,10,17],[3,22]],[24,[12,18,15,13],[3,23]],[21,[6,16,15],[2,25]],[22,[12,18,8,21],[3,25]],[24,[12,18,21,10],[3,19]],[21,[6,22,3],[2,24]],[22,[12,18,14,9],[3,18]],[24,[8,12,18,25],[3,32]],[21,[8,10,19],[2,27]],[22,[16,24,6,17],[3,20]],[24,[8,12,22,21],[3,29]],[21,[12,15,4],[2,16]],[22,[16,24,10,5],[3,15]],[24,[12,18,33,4],[3,17]],[21,[8,14,7],[2,21]],[22,[20,30,6,9],[3,14]],[24,[18,27,6,13],[3,21]],[21,[12,18,9,13],[3,23]],[22,[20,30,4,17],[3,19]],[24,[12,18,10,21],[3,26]],[21,[16,24,12,7],[3,16]],[22,[30,45,6,2],[3,11]],[24,[12,18,14,13],[3,20]],[21,[20,30,15,4],[3,13]],[22,[12,30,15,5],[3,14]],[24,[12,18,16,9],[3,20]],[21,[12,18,15,10],[3,19]],[22,[8,20,14,17],[3,23]],[24,[16,24,6,21],[3,24]],[21,[8,12,14,23],[3,29]],[22,[8,20,18,13],[3,21]],[24,[20,30,6,13],[3,17]],[21,[12,18,21,7],[3,16]],[22,[12,30,10,5],[3,14]],[24,[30,45,6,4],[3,13]],[21,[8,12,18,19],[3,26]],[22,[18,45,6,2],[3,12]],[24,[8,20,14,21],[3,27]],[21,[12,18,27,4],[3,15]],[22,[12,20,10,5],[3,14]],[24,[12,30,21,4],[3,15]],[21,[8,12,22,15],[3,23]],[22,[12,20,6,9],[3,15]],[24,[8,20,18,17],[3,24]],D

total :

2002/11/5(20:33)

CONSTRUCTION OF AFFINE PLANE CURVES

21

O

N

OT co
py
D
U for
PL
IC pr
AT oo
E frea
TH d
IS

CO

PY

[24,[18,45,6,4],[3,14]],[26,[14,35,5],[2,21]],[27,[8,12,22,27],[3,35]],[24,[8,28,18,9],[3,20]],[26,[10,45,4],[2,21]],[27,[18,27,6,16],[3,26]],[24,[12,16,14,13],[3,18]],[26,[4,22,33],[2,45]],[27,[12,18,10,27],[3,32]],[24,[12,16,18,9],[3,18]],[26,[4,26,29],[2,43]],[27,[18,27,12,10],[3,18]],[24,[16,24,12,18,9],[4,16]], [26,[4,30,25],[2,41]],[27,[12,18,14,19],[3,25]],[24,[16,24,12,14,13],[4,16]],[26,[4,34,21],[2,39]],[27,[12,18,16,15],[3,23]],[25,[2,51],[1,77]],[26,[6,14,29],[2,38]],[27,[16,24,10,15],[3,20]],[25,[3,26],[1,53]],[26,[6,16,25],[2,35]],[27,[20,30,6,19],[3,21]],[25,[6,11],[1,41]],[26,[8,10,29],[2,37]],[27,[28,42,6,3],[3,15]],[25,[8,12,15],[2,37]],[26,[10,14,5],[2,24]],[27,[12,30,15,10],[3,19]],[25,[10,15,11],[2,33]],[26,[20,30,25,4],[3,13]],[27,[8,20,14,27],[3,33]],[25,[18,27,5],[2,25]],[26,[8,12,18,29],[3,36]],[27,[12,30,21,7],[3,17]],[25,[26,39,3],[2,22]],[26,[8,12,22,25],[3,33]],[27,[8,20,18,23],[3,30]],[25,[34,51,2],[2,21]],[26,[30,45,10,4],[3,14]],[27,[12,30,27,4],[3,17]],[25,[6,15,20],[2,38]],[26,[12,18,10,25],[3,30]],[27,[8,20,22,19],[3,27]],[25,[6,21,17],[2,34]],[26,[16,24,10,13],[3,19]],[27,[12,30,10,15],[3,19]],[25,[10,35,6],[2,21]],[26,[8,20,14,25],[3,31]],[27,[12,30,14,7],[3,18]],[25,[4,18,35],[2,45]],[26,[8,20,18,21],[3,28]],[27,[12,30,8,19],[3,22]],[25,[6,27,14],[2,30]],[26,[8,20,26,13],[3,24]],[27,[8,28,14,19],[3,25]],[25,[4,22,31],[2,43]],[26,[8,28,18,13],[3,22]],[27,[8,28,22,11],[3,22]],[25,[6,33,11],[2,27]],[26,[16,24,20,10,13],[4,16]],[27,[12,16,14,19],[3,23]],[25,[4,26,27],[2,41]],[27,[2,55],[1,83]],[27,[12,16,22,11],[3,20]],[25,[6,39,8],[2,27]],[27,[3,28],[1,57]],[27,[12,16,18,15],[3,21]],[25,[4,30,23],[2,39]],[27,[4,19],[1,49]],[27,[12,20,10,15],[3,19]],[25,[4,34,19],[2,37]],[27,[7,10],[1,43]],[27,[12,28,6,3],[3,17]],[25,[6,51,2],[2,28]],[27,[10,15,12],[2,36]],[27,[16,24,12,18,15],[4,19]],[25,[9,12,17],[2,32]],[27,[28,42,3],[2,23]],[27,[16,24,12,14,19],[4,21]],[25,[15,20,6],[2,18]],[27,[6,15,22],[2,41]],[27,[16,24,12,22,11],[4,18]],[25,[9,15,14],[2,27]],[27,[12,30,7],[2,24]],[27,[16,24,20,10,15],[4,17]],[25,[6,14,27],[2,36]],[27,[22,55,2],[2,18]],[28,[2,57],[1,86]],[25,[6,16,23],[2,33]],[27,[6,21,19],[2,37]],[28,[3,29],[1,59]],[25,[6,20,15],[2,28]],[27,[8,28,11],[2,28]],[28,[8,9],[1,44]],[25,[6,22,11],[2,27]],[27,[10,35,7],[2,23]],[28,[8,12,17],[2,41]],[25,[6,26,3],[2,28]],[27,[6,27,16],[2,33]],[28,[14,21,8],[2,32]],[25,[8,10,27],[2,35]],[27,[4,22,35],[2,47]],[28,[38,57,2],[2,23]],[25,[12,15,8],[2,20]],[27,[6,33,13],[2,30]],[28,[6,15,23],[2,43]],[25,[8,14,15],[2,26]],[27,[10,55,2],[2,21]],[28,[6,21,20],[2,39]],[25,[10,12,11],[2,24]],[27,[4,26,31],[2,45]],[28,[6,27,17],[2,35]],[25,[12,18,9,17],[3,30]],[27,[6,39,10],[2,29]],[28,[8,36,9],[2,26]],[25,[20,30,15,6],[3,16]],[27,[4,30,27],[2,43]],[28,[4,22,37],[2,49]],[25,[12,18,15,14],[3,25]],[27,[4,34,23],[2,41]],[28,[6,33,14],[2,31]],[25,[12,18,21,11],[3,21]],[27,[6,51,4],[2,30]],[28,[4,26,33],[2,47]],[25,[8,12,18,27],[3,34]],[27,[4,38,19],[2,39]],[28,[4,30,29],[2,45]],[25,[12,18,27,8],[3,18]],[27,[9,12,19],[2,35]],[28,[6,45,8],[2,30]],[25,[8,12,22,23],[3,31]],[27,[12,16,11],[2,26]],[28,[4,34,25],[2,43]],[25,[18,27,6,14],[3,23]],[27,[15,20,7],[2,20]],[28,[4,38,21],[2,41]],[25,[12,18,10,23],[3,28]],[27,[21,28,3],[2,15]],[28,[6,57,2],[2,31]],[25,[18,27,15,5],[3,14]],[27,[9,15,16],[2,30]],[28,[9,12,20],[2,37]],[25,[18,27,12,8],[3,15]],[27,[6,14,31],[2,40]],[28,[9,15,17],[2,32]],[25,[12,18,14,15],[3,22]],[27,[9,21,10],[2,23]],[28,[12,20,9],[2,22]],[25,[16,24,6,23],[3,26]],[27,[12,28,3],[2,18]],[28,[6,14,33],[2,42]],[25,[20,30,6,15],[3,18]],[27,[6,16,27],[2,37]],[28,[6,16,29],[2,39]],[25,[12,30,15,8],[3,17]],[27,[6,20,19],[2,31]],[28,[9,24,8],[2,22]],[25,[8,20,14,23],[3,29]],[27,[6,22,15],[2,30]],[28,[6,20,21],[2,33]],[25,[8,20,18,19],[3,26]],[27,[6,28,3],[2,30]],[28,[6,22,17],[2,31]],[25,[8,20,22,15],[3,24]],[27,[8,10,31],[2,39]],[28,[6,26,9],[2,30]],[25,[12,30,10,11],[3,17]],[27,[12,15,10],[2,22]],[28,[8,10,33],[2,41]],[25,[12,30,8,15],[3,19]],[27,[8,14,19],[2,29]],[28,[8,14,21],[2,31]],[25,[8,28,14,15],[3,22]],[27,[10,12,15],[2,26]],[28,[8,18,9],[2,27]],[25,[12,16,14,15],[3,20]],[27,[10,14,7],[2,25]],[28,[10,12,17],[2,28]],[25,[12,20,10,11],[3,17]],[27,[16,24,12,11],[3,24]],[28,[12,18,15,17],[3,30]],[25,[16,24,12,14,15],[4,18]],[27,[20,30,15,7],[3,18]],[28,[16,24,20,9],[3,20]],[26,[2,53],[1,80]],[27,[28,42,21,3],[3,13]],[28,[12,18,21,14],[3,25]],[26,[5,14],[1,44]],[27,[12,18,15,16],[3,28]],[28,[8,12,18,33],[3,40]],[26,[22,33,4],[2,24]],[27,[12,18,21,13],[3,24]],[28,[8,12,22,29],[3,37]],[26,[8,20,13],[2,32]],[27,[8,12,18,31],[3,38]],[28,[12,18,33,8],[3,20]],[26,[10,25,9],[2,27]],[27,[12,18,27,10],[3,21]],[28,[18,27,6,17],[3,28]],

D

total :

2002/11/5(20:33)

22

M. FUJIMOTO AND M. SUZUKI

O

N

OT co
py
D
U for
PL
IC pr
AT oo
E frea
TH d
IS

CO

PY

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total :

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Mitsushi Fujimoto
Department of Mathematics
Fukuoka University of Education
Munakata, Fukuoka 811-4192, Japane-mail: fujimoto@fukuoka-edu.ac.jp
Masakazu Suzuki
Faculty of Mathematics
Kyushu University
36, Fukuoka 812-8581, Japane-mail: suzuki@math.kyushu-u.ac.jp