Automorphisms of the polynomial ring in two variables

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Moscow a ha ime, o Maka -Limano e e s o he esul as Semina gi en a Uni e si a Au ónoma de Ba celona, July 1981 . 155 unpublished wo k o Sha a e i ch .)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 . 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