ON THE JACOBIAN CONJECTURE

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1 v v v Far East Journal of Mathematcal Scences (FJMS) 17 Pushpa Publshng House, Allahabad, Inda Volume 11, Number 11, 17, Pages ISSN: ON THE JACOBIAN CONJECTURE 1-9 Kurauen-ban, Nshnomya Hyogo 66-8, Japan e-mal: fwh5864@nfty.com Abstract We show that the Jacoban conecture of the two dmensonal case s true. 1. Introducton Let T = ( P( x, y), Q( x, y) ): C ( x, y) C ( ξ, η) be a polynomal where c s a non-zero constant. Receved: January 19, 17; Accepted: Aprl 3, 17 1 Mathematcs Subect Classfcaton: 14R15. Keywords and phrases: Jacoban conecture, polynomal automorphsms. mappng. If T Aut( C ), ts Jacoban J ( T ) s a non-vanshng polynomal. Then J ( T ) s a non-zero constant. The Jacoban conecture says that f J ( T ) s a non-zero constant, then T Aut( C ). In 1993, Kalman showed that to prove the Jacoban conecture, we may assume that every level curve of P s rreducble [8]. In other words, there s a polynomal automorphsm S such that all level curves of P ( S) are rreducble when T = ( P, Q) s a polynomal mappng wth J ( T ) = c,

2 566 In 11, we clamed that the Jacoban conecture s affrmatve by usng the Kalman s result essentally but ncompletely []. In ths paper, we wll prove completely by the same method n a lttle refned manner.. Prelmnares and Defntons Defnton.1. We say that P ( x, y) s nonsngular f { P P = } x = y = and that a nonsngular polynomal P ( x, y) satsfes the condton (K) when ts every level curve s rreducble. Remar.. P ( x, y) = x( xy 1) s a nonsngular polynomal but { P = } s reducble [8]. Defnton.3. Let P ( x, y) satsfy the condton (K). We consder the followng lnear partal dfferental equaton where u ( x, y) s an unnown functon J ( P, u) = g( x, y). (1) We say that the problem (A) s solvable when for every entre functon g ( x, y), equaton (1) has a global soluton. Remar.4. We remar that the problem (A) s not solvable when some level curve s not smply connected (Lemma 5.1). So f the problem (A) s solvable, every level curve of P s smply connected and then the Jacoban conecture follows rather easly (Theorem 5.3). Conversely, f every level curve of P s smply connected, then the problem (A) s solvable (Proposton 4.7). Remar.5. In 1994, Bartolo et al. [3] showed that there s a polynomal whch satsfes the condton (K) whose level curve s not smply connected. Such a concrete example was shown n [5]. So the problem (A) s not solvable always.

3 On the Jacoban Conecture 567 Defnton.6. Let P ( x, y) satsfy the condton (K). When there s a polynomal Q ( x, y) such as J ( P, Q) = c, where c s a non-zero constant, we say that P satsfes the condton (J) and Q s an assocated polynomal wth P. Remar.7. When the problem (A) s solvable, P satsfes the condton (J) from Corollary of Theorem 4 n [7], because P satsfes the condton (K) and every level curve of P s smply connected (Remar.4). We wll solve the problem (A) under the condton (J). 3. Basc Facts Let P ( x, y) be a nonsngular polynomal. Followng proposton s a summarzed statement from Suzu [1, p. 4]. Proposton 3.1. Let P ( x, y) be a nonsngular polynomal. Then except for at most fnte number of level curves, every level curve { P = a} s rreducble and the same topologcal type ( g, n), that s, the normalzaton of { P = a} has a genus g and n punctured boundares. Remar 3.. If P satsfes the condton (K) and g = n the same notaton of above proposton, every level curve s smply connected from Theorem A n [9] and then the problem (A) s solvable. So almost all level curves of polynomal n Remar.5 have genus g > and no one s smply connected from Corollary of Theorem 4 n [1]. It s easy to see the followng proposton from Proposton 3.1. Proposton 3.3. Let P ( x, y) satsfy the condton (K). Let { P = a } be an exceptonal level curve n the sense of Proposton 3.1 and exceptonal values be { a } =. When a a,..., a, there s a suffcently small ds 1,..., 1 U ( a) wth the center a wthout a 1,..., a such that E a := {( x, y) ; U( a) P ( x, y)} s consdered topologcally as U ( a) R, where R s a ( g, n) -type Remann surface.

4 568 Proposton 3.4. Let P satsfy the condton (J) and Q s an assocated polynomal wth P. For every a C f we tae a suffcently small ds U ( a), we can consder E a : = {( x, y) ; U ( a) P( x, y) } as a fber space such as Ea = ( Ea, P, U( a) ) and there s a global holomorphc secton σ a of such as { Q( x y) = r} I E, where { Q ( x, y) = r} = σ s rreducble n, a and P taes every value b C on σ. Proof. Snce Q s a nonsngular polynomal, almost all level curves are rreducble by Proposton 3.1. Snce T = ( P, Q) s a locally bholomorphc mappng and the mage of T s a Zars open set, the complement of the mage of T contans at most fnte ponts of C ( ξ, η). So, t s easy to see E a C that there s a complex number r such as above. Remar 3.5. Note that σ a n Proposton 3.4 s smply connected. When a b, σa I σb = or σa I σb s a smply connected open set on σ. 4. Solve the Problem (A) We wll solve the problem (A) under the condton (J). To construct a global soluton of (1), we prepare lemmas about local soluton. Snce (1) s a nonsngular lnear partal dfferental equaton, there s a unque (sngle valued) holomorphc soluton u a near σ a havng any gven holomorphc ntal data on σ a by Cauchy-Kowalevsaya s theorem and the assumpton that σ a s smply connected. When we restrct equaton (1) to { P = b}, where b U( a), t s a g g holomorphc 1-form such as du = dx = dy, where P dx P dy Py P x + y x =. In other words, { P = b} s a characterstc curve of (1). So above local soluton u a can analytcally prolong along any path on { P = b}.

5 Lemma 4.1. Let On the Jacoban Conecture 569 E a be same n Proposton 3.3. Then above soluton u a can prolong free n E a, that s, u a can (analytcally) prolong along any path from a pont p of σ to a pont q n E. a Proof. We remar that the unversal coverng space E ~ a can be ~ consdered as U ( a) R. Snce u a can be consdered as a sngle valued functon on E ~ a and holomorphc near σ a, u a s holomorphc on E ~ a by Hartogs theorem. It means that u a can prolong free n E a. Lemma 4.. We use the same notatons n above lemma. Let u be a functon element whch s obtaned by an analytc prolongaton of u a along a path l from a pont p σa to a pont q where l Ea and q { P = b}. Set σ I { P = b} = p. Then there s a path l from p to q where a l { P = b} such that we have a functon element u when we prolong u a along l. Proof. From the proof of Lemma 4.1, u a can extend a holomorphc ~ functon on E a. Let ~ l be the lft of l and proect t to { P = b} as l. It s easy to see that we have a functon element u when we prolong u a along l. Lemma 4.3. Let a Ea, Eb, a b be same as Proposton 3.3 and σa I σb. If u a = ub on σa I σ b, then ua U ub can prolong free n Ea U E b. Proof. We have a concluson easly from Lemma 4.. From Lemmas 4.1 and 4.3, followng corollary s obtaned easly. Corollary 4.4. Let a be an exceptonal value and U ( a ) be a ds wth the center a wthout other exceptonal values. Let U ( a ) = { U( a ) a } and Ea = { U ( a ) P( x, y)}. Then u a can prolong free n ( a ). E

6 57 Lemma 4.5. Let a and U ( a ) be the same n above corollary. Then can prolong free n E a, where Ea = { U( a ) P( x, y) }. Proof. By Tuber Neghborhood Theorem (see [6, p. 67], for example), there s an open Sten neghborhood U of S = { P = α } n E α and a homotopy of holomorphc map f s : U U, s [, 1] such that f s the dentty map on U, f s S s the dentty map on S for all s [, 1], and f ( U ) =. 1 S At frst, let l be an arbtrary path from p to q S such as l U and u be a functon element at p whch satsfes equaton (1). When we consder f s (), l they are homotopy curves such that f ( l) = l, f1 ( l) S and these termnal ponts are q. So u can prolong along the path l snce S s a characterstc curve and ts functon element at q satsfes equaton (1) from the prncple of permanence of functonal relatons. At second, let l be a path from dscusson and Corollary 4.4, Fnally, let l be an arbtrary path n u a p σ a to q S n E. From above u a can prolong to q along l. E a from above dscusson and Corollary 4.4, t s easy to see that a p σ a to q. Then from u a can prolong along l to q. Theorem 4.6. When P satsfes the condton (J), the problem (A) s always solvable. Proof. Let σ be a countable coverng of σ and ts proecton by P s U such as U ( a) n Lemma 4.1 or U ( ) n Corollary 4.4. And we use notatons ntal data on a E smlarly as Lemmas 4.1 and 4.5. Let u be a holomorphc σ and u be all functon elements obtaned by the prolongaton along every path n E havng the ntal data u.

7 On the Jacoban Conecture 571 When σ I σ = σ, s [, 1] on σ I σ s a holomorphc functon near b U I U, σ and J ( u u ) =. Then t s easy to see that for every u u s a constant on { P = b} and we can defne O ( σ I ) such as u u = ϕ ( P) near σ. σ σ l. σ From the defnton, t s easy to see that ϕ =. And when ϕ I σ I σ = σ, t s easy to see that ϕ + ϕ + ϕ = l l ϕ l l on Then { ϕ : σ I σ} s a Cousn-I data. Snce σ s a Sten manfold, one can fnd functons { ϕ O ( σ )} such as ϕ ϕ = ϕ when σ I σ. We set u = u + ϕ ( P). It prolongs free n E. Let E I E and σ I σ = σ. It s easy to see that u + ϕ ( P) ( u + ϕ ( P)) = on σ. And then u U u can prolong free n E U E. By the successve dscusson, and t s a global soluton by the monodromy theorem. U =1 u prolongs free n C = U = 1E Proposton 4.7. When P satsfes the condton (K) and every level curve s smply connected, the problem (A) s solvable. Proof. Let us recall Theorem.1 n [1]. To prove above proposton, we must chec the condton (a) and (c) n the statements of Theorem.1. From the condton (K), t s easy to see that the condtons (a) and (c) are satsfed. 5. Concluson Lemma 5.1. If the problem (A) s solvable, then { P = a} s smply connected for every a C.

8 57 Proof. If some level curve S = { P = a} s not smply connected, then P there s a holomorphc 1-form a ( x) dx( = b( y) dy), where ( ) = y b y a( x) P x whose ntegral on S s a mult-valued functon by Behne-Sten theorem [4]. If we set a( x) P = b( y), then t represents a holomorphc functon g y P x on S. Snce C s a Sten doman, there s an entre functon g ( x, y) such g that g S = g. From the explanaton n outset of Secton 4, du = dx = P a ( x)dx on S. It s easy to see that equaton (1) for that g has no global y soluton. Ths contradcts the assumpton. Next proposton s a specal case of Theorem 3.3 n [1]. Proposton 5.. Let P ( x, y) satsfy the condton (K) and every level curve s smply connected and u ( x, y) be an entre functon. Then ( P, u) Aut( C ) f and only f u satsfes the equaton J ( P, u) = ϕ( P) where ϕ s an entre functon such as ϕ. Theorem 5.3. If the polynomal map T = ( P( x, y), Q( x, y) ) satsfes J ( P, Q), then T Aut( C ). Proof. By vrtue of Kalman, we may assume that P satsfes the condton (K). From Theorem 4.6, Lemma 5.1 and Proposton 5., we conclude T Aut( C ). Followng proposton s an explct verson of Jung [11]. Proposton 5.4. If ( P, Q) ( C ), P = x + ψ( cy + ϕ( x) ), Q = Aut alg cy + ϕ( x), where ϕ, ψ are polynomals of one complex varable and c s a non-zero constant. And the nverse s true.

9 On the Jacoban Conecture 573 Proof. Let ( P, Q), ( P, Q ) ( C ) such as J ( P, Q) = J ( P, Q ) = c. Aut alg Then J ( P, Q Q) =. As the same reason of the proof of Theorem 4.6, Q Q = ϕ( P), where ϕ s a polynomal of one complex varable. Let ( P, Q ), ( P, Q ) ( C ) such as J ( P, Q ) = J ( P, Q ) = c. Aut alg Then smlarly as above, P P = ψ( Q ), where ψ s a polynomal of one complex varable. Let J ( P, Q ) ( C ) such that J ( P, Q ) = c. Aut alg Then P = P + ψ( Q ), Q = Q + ϕ( P). Now we set P = x, Q = cy. Then we have above the concluson. Let P = x + ψ( cy + ϕ( x) ), Q = cy + ϕ( x), where c s a non-zero constant. Then J ( P, Q) = c. Also, ( P, Q) ( C ). Aut alg References [1] Y. Adach, Condton for global exstence of holomorphc solutons of a certan dfferental equaton on a Sten doman of Japan 53 (1), n+1 C and ts applcaton, J. Math. Soc. [] Y. Adach, On the Jacoban conecture, Poneer J. Math. Math. Sc. (11), [3] E. Artal-Bartolo, P. Cassou-Noguès and I. Luengo Velasco, On polynomals whose fbers are rreducble wth no crtcal ponts, Math. Ann. 99 (1994), [4] H. Behne and K. Sten, Entwclung analytsher Funtonen auf Remannschen Flächen, Math. Ann. 1 (1949), [5] A. van den Essen, Polynomal automorphsms and the Jacoban conecture, Progress n Mathematcs, 19, Brhäuser Verlag, Basel,. [6] F. Forstnerč, Sten Manfolds and Holomorphc Mappngs, Sprnger, 11. [7] O. Futa, Sur les systèmes de fonctons holomorphes de plusers varables complexes (I), J. Math. Kyoto Unv. 19 (1979), [8] S. Kalman, On the Jacoban conecture, Proc. Amer. Math. Soc. 117 (1993),

10 574 [9] H. Satō, Fonctons entères qu se rédusent à certans polynomes (II), Osaa J. Math. 14 (1977), [1] M. Suzu, Proprétés topologques des polynômes de deux varables complexes, et automorphsmes algébrques de l espace C, J. Math. Soc. Japan 6 (1974), [11] H. W. E. Jung, Über ganze bratonal transformatonen der Ebene, J. Rene Agnew. Math. 184 (194),