Pub
.
Ma
.
UAB
Vol
.
27
Ne
1
AUTOMORPHISMS
OF
THE
POLYNOMIAL
RING
IN
TWO
VARIABLES`
Wa en
Dicks
Le
k be a
ield, k[x,y]
he
polynomíal
ing
in
wo
a iables,
and
Au
k[x,y]
he
g oup
o
all
i s
k-algeb aau omo phisms
.
Such an
au o-
mo phísm
will
be
deno ed
by
he
o de ed
pai
(p,q)
whe e
p,q
E
k[x,y]a e
he
espec i e
images
o
x,y
.
THEOREM
.
The
g oup
Au
k[x,y]
is
gene a ed
by
(y,x),
(x,y
-uxn
)
u
E
k,
n
;
0
.
Mo eo e
Au k[x,y]
=
A*
U
B
whe e
A
=
{(X11x+ñ12y+ñ1'X21x+a22y+ñ2)IX11x22#~21>~12},
B =
{(a11x+Xl,a22y+ (x))IXlla22#O, (x)
E
k[x]},
0 =
An
B = {(X 11
x+a1',
21
x+a22y
+a
2)I
a
11
a
22~0}
.
The
elemen s
o
A
a e
called
a ine
au omo phisms,
he
elemen s
o
B
de
Jonquil es
au omo phisms,
and he
elemen s
o
he
subg oup
gene a ed
by
AUB
a e called
ame
au omo phisms
.
The
ac
ha
all
k-algeb a
au o
mo phisms
o
k[x,y]a e
ame
was
p o edby Jung
[2]
o
cha
k
= 0,
and
hen
by
Van
de
Kulk
[8]
in
he
gene al
case
.
F om
hei
wo k
he
cop oduc
decomposi ion
ollows
ai ly
easily,
bu
i
is
no
clea
who
i s
made
he
obse a ion
.
(Kambayashi
[3]
gi es
he
c edi
o
Sha a e i ch
[7]
.)
Ren schle
[5]
ga e
a
e y simple
p oo
o ameness
o
cha
k
=
0,
and
hen
along
sligh ly
di e en
lines
Maka -Límano
[4]
ga e
a
ai ly simple
p oo
o
a bi a y
cha ac e ís ic
.

(News
o
Van
de
Kulk's esul
seems
no
o
ha e
eached
Moscow
a
ha
ime,
o
Maka -Limano
e e s
o
he
esul
as
Semina
gi en
a
Uni e si a Au ónoma
de
Ba celona,
July
1981
.
155
unpublished
wo k
o
Sha a e i ch
.)
In
Che
spi i o
Se e
[6l,
Roge
Alpe in
El]
ga e an
explici
example
o
a
ee
ac ed
on by
Au k[x,yl
om
which
Che
cop oduc
decomposi ion
can
be ead
o
.
In
§l
below
we
gi e
a
modi ied
e sion
o
Maka -Limano 's
p oo ,
and
in
§2
ecall
Alpe in's
example
.
I
am
e y
g a e ul
o
P .M
.
Cohn
o
p o idíng
me
wi h
his
ansla ion
o
Maka -Límano 's
hesís
.
§l

The
suppo
o
a
p imi i e
elemen
Le
( ,g)
be an
au omo phism
o
k[x,yl
.

Ide
can
w i e
=
Ix
i
jx
l
y
j
,
aij
e
k
and
de ine
supp( )
_ {x y
l
Iaij
¢ 0} _
<x,y>,
whe e
<x,y>
is Che
ee
abelían
g oup
gene a ed
by
x,y
.
Le
m
=
x-deg( ),
n =
y-deg( ),
ha
is,
m
is
Che
highes
exponen
o
x
occu íng
in supp( ),
and
simila ly
o
n
.
Se
o = {x
l
y
j
lni+mj
<
mn,
i
? 0,
j
3
0}
S
<x,y>
.
Geome ically,
supp( )
lies
in Che
ec angle
de e mined
by
l,x
m
,x
m
yn,yn
and
A
occupies
Che
iangle
m
de e minedby
l,x~,y
n
.
The objec i e
o
Chis
sec ion
is o
show
x
m
,y
n
e
supp( )
c
Q
and
min
o
nlm
.
I mn
=
0
Chis is
clea
.
Thus we
may assume
mn
>'O
.
Le
m' =
m/(m,n),
n'
=
n/(m,n)
.
These
a e
cop imena u al
numbe s,
so we
can
choose
na u al
numbe s
s,
such
ha
sm'- n'
=
1
.
Le
u =
xm
/yn
,
=
y
s
/x
in
<x,y>
so
x
= us
n
,
y = u
m
.
15
6
y
s
/x
=
n
m' n'
u=x
/y
Thus k[x,y]
c
k[u, ]
and
we
can
w i e
£
=
S
ij
u1
]
so
supp( )
_
{ul ]Ipij
0}
.
We de ine
he
leading
-componen
o
o
be
I l
=
(~
u
.
.U'
)V
ie
k[u]
> '
x
< >
i
1J
whe e
j
=
-deg( )
.

I
hen
u-deg(I l)
=
i
we
de inejI ll
=ul
i
E
<u, >
called
he
leading
e m
o
.
This
ex ends
o a
g oup
homomo phism
11 II
:
k(u, )
x
->
<u, >
.

(No ice
he
supe sc íp
x
is
being
used
o
deno e
he
se
o
nonze o
elemen s
.)
The
ollowing
s a emen indíca es
he
s eps
in
Maka -Limano 'sa gumen
.
TRE
OREM
1
.

(í)

The eexis
a,s
E
k(u)
x x < > c
k(u, )x
such
ha
¡ ¡

=

aa
a
(a

E
k
x
,

a

EIN
+
)

and x,y

e
k[a
+l,
o]
.
(ii)
The e
hen
exis
w,z
E
<u, >
such ha
<w>
_
<~~II>
o
<IIaII
.
~Ia~I>
and
+l
x,y
e
semígp<w
,z>
.
(iíi)
Then
xm
,yn
E
supp( )
c p
and
¡¡ ¡¡
=
xm
and
<w> =
<x>
.
(í ) I
<w>
=
<Ilaib
hen
mln
.
( )

I <w> =
~II«II,
IISIb
hen
nlm
.
PROOF
.

(i)
Le
K =
k(u)
and
conside
he
Lau en
se ies
ield
K((
-1
))
.
In
a
na u al
way
k(u,
c
K_ (( -1
))
and
he e
a e
maps
-deg
:
K((
1
))
x
-~
7L,
1

1 :
K((
-1
))
->
Kx
x
< >
ex ending
he
co espondíng
maps on
k[u, ]
.

We
iew
k
x
as a
subg oup
o K
x
x
< >
c
K((
1
))
x
.

Since
-deg( )

>
0
he e
exis s
aE
K
x x
< >
such ha
he
ímage
o a
in
(Kx
x
< >)/k
x
gene a es
a
x
maximal
cyclic
subg oupcon aining
he
ímage
o
I l,
say
I ¡
= aaa
a
Ek
,
a
c
]N
.

By
índuc íon
on
a
we
shall
show
ha
o any ,g
e
K((
-1))
wi h
I
l
= aaa
a
E
k
x
,
aE
IN "
'
he e
exis s
R
E
K
x
x< >
such ha
Ik[
±l ~g]I c
k[a±l,o]
.
The
case
a =
0
is
acuous
.
Le
us
now de ine
a
(possibly
ini e)
sequence
induc i ely
.
Le
an
í
9, = g
.
Suppose
we ha e
gí
o
some
i
>,
1
.
I
Igil
=
a
l
a)

o
some
n
i
a
l
E
k',
ni
E
7i
we
se
gí+1
=
gi-xí
;
í
g
í = 0 o g
i
¢ 0
and
Ig
i
l
is
no
o
his
o m we
le
he sequence end
a
he
sequence g
1
9
2
,
. . .
has a
limi
g
*
in
K((
-1
)),
I
g
*
=
0
hen
k[ ±l,g]
c
k((
-l ))

so

Ik[
1
c
k[I l l]
c
k[
.±1]
and we
can
ake
R
a bi a y
.
Thus
we
may
assume g
* ¢
0
so
he sequence
is
ini e and
k[
±l
.
g]
l~
k[
±l .g
*
]
"
I
I 1,Ig*)
a e
algeb aically
independen
o e
k
hen
i is
easy
o
see
Ik[ ±l ,g
*
]
x
i
c
k[I I±1,Ig*I]
and
we
can
ake
S
=
Ig*I
.
Thís
lea es
he case
whe e
I j,Ig*I
a e
algeb aically
dependen
o e k
.
I
c
=
-deg( ),
d =
-deg(g
*
)
hen
I id,Ig*I, a e
algeb aicallydependen
o e k
and a e
-homogeneous
wi h he same
-deg ee
.
I
ollows
ha
I ld/ig*Ic
lies
in
K
and
is
algeb aic
o e k
so
lies
in k
.
Thus
i
Ig*I
c
-

I l
d-
aad
(mod
k
x
) .

Bu

(K
xx
< >)/k
x
is
a
o sion- ee
abelian
g oup,
and he ímage o a
gene a es
a
maximal
cyclic
subg oup,
so
ciad
and
Ig*I
=
a
b
(mod
k
x )
whe e
b =
ad/c
.
Saylg
*
1=
pab
,
p e kx
.
By
he
de ini ion
o g* we
know
alb,
say b
=
aq+

0< <a
.

Le h = g*
/
q
.
Then
Ihi
-
a
(mod
k
x )
and he
induc ion hypo hesis
applies
o
he pai
(h, )
.
_
Hence
he e
exis s
B s
Kx
x
< > such
ha

Ik[h +1
, ]
x
I
ck[a
±1
,B]
.

Now
Ik[
±1
,g]x
i

Ik[ +l,h]x1
cIk[ ,h]xI<I I> ck[a
l
,s]
.
By
induc ion
Ik[ ±l,g]xj
c
k[aS]
o some R
e
k(u)
x
x <
>,
and
(i)
is
p o ed Since
x
.y
e
Ik[ ,g]x
i
.
(ii)
Recall ha
wo
elemen s
o <u, > a e saíd
o
be
dependen
i
hey
gene a e
a
cyclic
subg oup,
and
o he wise
hey
a e independen , ha
is,
eely gene a e a
ee
abelian
subg oup
.
I
IIaII,
IISIIa e
independen
hen
i
is
clea
ha
x,y
e
IIk[a+l,s_

]x
1I
c

semigp<IIal~1,IISII>
and we can
ake
w =
¡¡al¡,
z
=11011
.
This lea es
he,
case
whe e
IIaII
,
IISIIa e
dependen
.

Le
w
be a
gene a o
o
11
.11, 1101¡
:
,
say 11-11 =
w
l
.
11611
=
w
j
,
w
=
IIai il0Ih
"
He e
Since
-deg(g
1
)
>
-deg(g
2
)
>
n
g*
= g-a
l
1-a
2
n2
g]
x
i,
Ik((
-1 ))
x
1
Ila~
i~
=
IIPlIl
=
w
lj
so
he e
is
a
uníque V
e
kxsuch
ha
z
=
Ila'-PPijj

w
íj
Bu
z
and
w
lj
ha e
he
same
-deg ee
so
w,z a e
independen
.
Le
a'
= acPd
.,
P'

= a
j
/P
1
-
u
.

Then
II
k[
a+l
,P
+1
7
x
,1
=
llk[a'
+l
.
(P'+u)+l]xll
c
Iik[a'
.P']
x
Il
<
w
>
c
semigp<w
±l
,z>
.
Thus
x,y
e
semigp<w
±
l,z>
and
<w> _
<
llalh
llPib
.
(iii)
Geome ically
x,y
e
semigp<w
±1,
z>
means ha one o he
wo
hal -planes
de e mined
by
w
con ains
bo h x
and
y
.
Ngw
by
(íi)
llall
=
w
l
o
some
in ege
i
and
on
eplacing
w
wi h
w
1
_i
necessa y
we
may
assume i 3 0
.

By
(i),
li
Il
=
h
i l
=wia
and
li
li
E

semigp<x,y>

so
wE

semigp<x,y>
.

The only
way'
his
can
happen
is
o
w
o
líe
along
he x
o
y axis,
ha
is,
w
ís
a
powe
o x
o
y
.
Bu <w,z> ? <x,y> so w
ís
x
o
y
.
Thus
ll ljis
a
powe
o x o y
.
Bu
he
only place supp( )
mee s
he
x
o
y axes
is
in
A
so
ll lle
Aand
his
(o ces supp( )
S A
.
The
only
way
x-deg( ) can
be
m
ís
o x
m
o be
ín
supp( ),
and
simila ly
yn
e
supp( )
.

Thus1I ll
= x
m
o y
n
.

Bu
u-deg(x
m
)
=
u-deg(u
ms
mn'
)
=
ms
.,
u-
deg(y
n
)
.
=
u-deg(un m'n)
= n
= ms-(m,n)
<ms

so
li
Il
=
x
m
.

Hence
<w>

=-<x>
.
(i )

I
~lall>

=
<
w
>

=<x>

hen
II
a
II
=x
.

Bu

by

(i)
ll ll
=
llaIi
a
=

x
a
and

by
(íii)ll ll
=x
m
so
a =
m
.

Thus
I l
=
aa
m
ín k(x,y)
so
y-deg(l 1)
=
m(y-deg(a))
.
And
y-degl l = n sínce yn EsupPI I, so
min
.
( )
I
<
Ilall,
IIPII>
_ <
w
>
=
<x>
hen
n'7l=
-deg(<x>)
=
-
deg«JIali,
IIPII'
.)
=
-deg(<a,P>,)
.
By
(i)
y
c
k[n
1
,P]
and
his
.is
-homogeneous
so
-deg(y) E
-deg(<a,P>),
ha
is,
-
m'
is
a
mul íple
o n' so
nim
.
.
§2

The
Au omo phi
sm
G oup
C
Fo
any
p =

ijx1yj

E
k[x,y]
x
,
we de ine
deg(p)
=
max{i+jluij
¢
0}
;

i
deg
p =d we
de ine
p
0
=
"id-íxiyd-i
called
he
leading
componen
o
p
.
THEOREM
2
([2],[8])
.
Le
(p,q)
be
a-k-algeb a
au omo phísm
o
k[x,y]
wi h
deg
p

:5
deg q
.

Then
. .
ei he

(p,
q)

ís
a ine
o
he e
is
a uníque y

E
kx

and
Posi i e
ín eSe
such ha
deg(q
-
uP
)
<deg(q)
"

PROOF
.
Le
( ,g)
be
he
in e se
o
(p,q)
and le
be as
in
§l
.
li
deg(p
m
)
¢
deg(g
n
)
hen
deg( (p,q))
=
max{deg(p
m
),deg(g
n)}
.
Bu (p,q)
= x
so
p o q is a
polynomial
in
x o
deg ee
1 and he
desi ed
conclusion
ollows
easily
.
This lea es
he
case
whe e
deg(p
m
)
=
deg(g
n
)
.
He e
m
>.
n so
nlm
and
deg(p
)
=
deg(q)
o
=
m'

We
may
assume
(p,q)
is
no
a ine
so
deg
q
>
1
.
n
Since
(p,q)
=
x i
ollows
ha
p0,g
0
a e
algeb aically
dependen
o e
k
.
Hence
q0/p o
is
algeb aic
o e
kso
lies
in k,
say
q
0
0
=
u
.
Then
deg(q
-up
)
<
deg(q) as
dési ed
.
By
induc ion
on
deg(q)
i
ollows
easily om
Theo em
2
ha
all
k-algeb a
au omo phisms
o
k[x,y]
a e
ame
.
I
is
e en
a
simple
ma e
o
ob ain
he
decomposi ion
.
THEOREM
3
.

Au
k[x,y]
=
A
*C B
.
PROOF
.
Le
P
be he
o ien edg aphwhose e ices
a e he
k-subspaces
o
k[x,y]
and
whose edges
a e he
inclusion
maps
.

Then
Au
k[x,y] ac s in
a
na u al_
way
on
_he
.
- aph
P_
Le
T
be
he
o bi
o
k+kx
3
1-1-1--
,
Í
.
L'--
elaim
ha
T is a
ee
.
Any e ex
o T is
o
he
o m k+kp o
k+kp+kq
whe e
(p,q) ís
some
au omo phísm
.
We
de ine
deg(k+kp)
=
deg(p) and
deg(k+kp+kq)
_
max{deg(p),deg(q)}
-
} .
I is
easy
o
see
hese
a e well-de ined
.
Conside
a
e ex
o
he
o m k+kp
.
We
can
ind an
au omo phísm
(p,q)
wi h
deg(q)
minimal,
so
deg(q)
<
deg(p) o
(p,q) is
a ine
.
All
he
neighbou s
o
k+kp
a e
o
he
o m
k+kp+k(q+h)
whe e
h
e
k[p]
.

The
only
neighbou
o
k+kp wi h
smalle
deg ee
is
k+kp+kq
;
all he
o he s ha e
g ea e
deg ee
.
Conside
a
e ex
o
he
o m
k+kp+kq
whe e
deg(q)
<
deg(p)
.

The
neighbou s
a e
o
he
o m
k+k(ap+Sq)
whe e
a,R
e
kX
a e
no
bo h
ze o
;
only
k+kq
has
smalle
deg ee,
all
he
o he s ha e
g ea e
deg ee
.
Finally,
he
e ex
k+kx+ky
has
smalle
deg ee han
all i s
neighbou s
.
Thus
e e y
pa h om
k+kx+ky
is
s ic ly
inc easing
(so
T
has
no
ci cui s)
and
om
each
e ex
he e
is
a
s íc lydec easing
pa h
which
mus
necessa ily
a i e
a
k+kx+ky
(so T
is connec ed)
.
Hence
T is a
ee
.
Now
k+kx
;
k+kx+ky
is a
ans e sal
in
T
o
he
ac iono
Au k[x,y]
and he
s abilize
o
k+kx
ís B
while
he
s abilize_o
k+kx+ky
ís
A
.
Tb'
.s
implies
G =
A
*c B
.
c_°
C6]
.
REFERENCES
1
.
R
.C
.
ALPERIN,
Homology
o
he
g oup
o
au omo phísms
o
k[x,y],
J
.
Pu e
and
Appl
.
Algeb a
15
(1979)
109-115
.
2
.
H
.W .E
.
JUNG,
Ube
ganzebi a ionale ans o ma ionen
de
Ebene,
J
.
eine
angew
.
Ma h
.
184
(1942)
161-174
.
3
.
T
.
KAMBAYASHI,
On
he
absence
o
non i ial
sepa able
o ms
on
he
a ine
plane,
J
.
Algeb a
35
(1975)
449-456
.
4
.
L
.G
.
MAKAR-LIMANOV,
On
au omo phisms
o
ce ain
algeb as
(Russian),
Ph
.D
.
Thesís,
Moscow,
1970
.
5
.
R
.
RENTSCHLER,
Opé a ions
du g oupe
addi i
su le
plan
a íne,
C .R
.
Acad
.
Sc
.
Pa is,
Se
.A,
267
(1968)
384-387
.
6
.
J .-P
.
SERRÉ,
A b es,
amalgames
e
SL
2
,
As é ísque
No
.46,
Socié é
Ma h
.
de
F ance,
1977
.
7
.
I
.R
.
SHAFAREVITCH,
On some in íní e
dimensional
g oups,
pp
.208-212,
A í-Simposío
In ecnaz
.
di Geom
.
Alg
.
Roma,
1965
.
8
.
W
.
VAN
DER
KULK,
On polynomial
ings
in
wo
a iables,
Nieuw
A chie
oo
Wisk
.
(3) I
(1953)
33-41
.
Rebu
el
15juliol
1982
Re isa
el
20
ab il
1982
Bed o dCollege
Regen 's
Pa k
London,
NW1 4NS