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JOURNAL OF ALGEBRA ARTICLE NO.

Remarks on the Jacobian Conjecture Jie-Tai YuU Department of Mathematics, The Uni¨ ersity of Hong Kong, Pokfulam Road, Hong Kong Communicated by Walter Feit Received March 21, 1996

Let F [ Ž F1 , . . . , Fn . g ŽCw X1 , . . . , X n x. n with detŽ J Ž F .. g CU and let Mi Ž X i , Y . s m i0 Ž Y . q m i1Ž Y . X i q ??? qm i d i Ž Y . X id i g Cw X i , Y x [ C w X i , Y1 , . . . , Yn x be the minimal polynomial of F over CŽ X i .. We prove that m i0 Ž Y ., . . . , m i d i Ž Y . have no common zeros in C n. As a direct consequence, we obtain flatness of Cw F, X i x over Cw F x for every i. As applications, we obtain simple algebraic proofs of the following two known results: Ži. A birational polynomial map from C n into C n with detŽ J Ž F .. g CU is actually an automorphism; Žii. an injective polynomial map from C n into C n is also an automorphism. Q 1997 Academic Press

1. INTRODUCTION The main objective of this article is to prove the following new result. THEOREM 1. and let

Let F g ŽCw X x. n [ ŽCw X 1 , . . . , X n x. n with detŽ J Ž F .. g CU

Mi Ž X i , Y1 , . . . , Yn . s m i0 Ž Y . q m i1 Ž Y . X i q ??? qm i d iŽ Y . X id i be the minimal polynomial of F1 , . . . , Fn m i0 Ž Y ., . . . , m i d iŽ Y . ha¨ e no common zeros in C n.

o¨ er

C Ž X i .. Then

As a direct consequence, we obtain another new result. THEOREM 2. Let F g ŽCw X x. n [ ŽCw X 1 , . . . , X n x. n with detŽ J Ž F .. g C . Then Cw F, X i x is flat o¨ er Cw F x for e¨ ery i. U

As applications of Theorem 1, we obtain simple algebraic proofs of the following two known results. U

E-mail: [email protected] Partially supported by RGC-Fundable Grant 344r024r0001. 90

0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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THEOREM 3. Let F s Ž F1 , . . . , Fn . g ŽCw X x. n s ŽCw X 1 , . . . , X n x. n : C n ª C n be a polynomial map with detŽ J Ž F .. g CU , such that CŽ F . s CŽ X .. Then F is an automorphism; that is, Cw F x s Cw X x. THEOREM 4. Let F s Ž F1 , . . . , Fn . g ŽCw X x. n s ŽCw X 1 , . . . , X n x. n : C n ª C n be injecti¨ e. Then F is an automorphism. Theorem 1 and Theorem 2 may be viewed as generalization of two theorems of Formanek in w4x. Theorem 3 was first proved by Keller w5x in 1939, as the first step toward a confirmation of a conjecture he formulated, now called the Jacobian conjecture, which claims that if F: C n ª C n is a polynomial map with detŽ J Ž F .. g CU , then F is an automorphism. To our best knowledge, this conjecture is still open for n G 2 Žthe case n s 1 is trivially true.. For a history and related topics, see Bass et al. w3x. Keller’s original proof uses deep algebraic geometry which is not easily accessible. See also Bass et al. w3x for an alternative proof by subtle commutative algebra. We believe that the algebraic proof of Theorem 2 given here is simpler, since it only uses the notions of minimal polynomials and of Zariski closures from elementary algebraic geometry. Theorem 4 was first proved in w2x. Recently Rudin has published a simple proof in w7x by complex analysis. It seems the alternative proof obtained here for Theorem 3 has a somewhat more algebraic flavor. Notation. We often use X, Y, and F to denote Ž X 1 , . . . , X n ., Ž Y1 , . . . , Yn ., and Ž F1 , . . . , Fn ., respectively.

2. PRELIMINARIES LEMMA 5. Let f 1 , . . . , f n4 ; K w X 1 , . . . , X ny1 x contain n y 1 algebraically independent polynomials o¨ er the ground field K. Then there exists a unique Ž up to a factor from K U . irreducible polynomial GŽ Y1 , . . . , Yn . g K w Y1 , . . . , Yn x such that GŽ f 1 , . . . , f n . s 0. G is called the minimal polynomial of f 1 , . . . , f n o¨ er K. G has the following property: if L g K w Y x is such that LŽ f 1 , . . . , f n . s 0, then G < L in K w Y x. Moreo¨ er, GŽ Y1 , . . . , Yn . is the generator of the principal ideal

Ž ² f 1 y Y1 , . . . , f n y Yn : K w X1 , . . . , X ny1 , Y x . l K w Y x of K w Y x. Furthermore, the minimal polynomial of f 1 , . . . , f n o¨ er K Ž K is the algebraic closure of K . is the minimal polynomial of f 1 , . . . , f n o¨ er K.

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Proof. See, for instance, Theorem 1 in w9x and Theorem 1 in w10x. LEMMA 6. Let f 1 , . . . , f n4 ; Cw X 1 , . . . , X ny1 , T x contain Ž n y 1. algebraically independent polynomials o¨ er CŽT . such that for e¨ ery t g C,

f 1Ž T s t . , . . . , f n Ž T s t . 4 ; C w X1 , . . . , X ny1 x contains Ž n y 1. algebraically independent polynomials o¨ er C. Then Ži. We can uniquely Ž up to a factor in CU . choose the minimal polynomial of f 1 , . . . , f n o¨ er CŽT . as an irreducible polynomial in Cw T, Y x, i.e., the generator GŽT, Y . of the principal ideal

Ž ² f 1 y Y1 , . . . , f n y Yn :C w X1 , . . . , X ny1 , T , Y x . l C w T , Y x of Cw T, Y x. Žii. For all but finitely many t g C, GŽ t, Y . g Cw Y x is irreducible in Cw Y x and GŽ t, Y . is the minimal polynomial of f 1ŽT s t ., . . . , f nŽT s t . o¨ er C. Žiii. For any t g C such that GŽ t, Y . is reducible in Cw Y x, the minimal polynomial of f 1ŽT s t ., . . . , f nŽT s t . o¨ er C di¨ ides GŽ t, Y . in Cw Y x. Proof. Ži. As in Theorem 1 in w9x: clear denominators and discard any factors that depend on T alone. If the resulting polynomial has a nontrivial factorization, so does the original one. Žii. By Lemma 5, GŽT, Y . is the minimal polynomial of f 1 , . . . , f n over the C Ž T . . Hence GŽT, Y . is irreducible in C Ž T . w Y x. By the Hilbert Irreducibility Theorem Žsee, for instance, w8x., GŽ t, Y . is irreducible in Cw Y x for all but finitely many t g C. By substitution, GŽ t, f 1ŽT s t ., . . . , f nŽT s t .. s 0. Žiii. GŽ t, f 1ŽT s t ., . . . , f nŽT s t .. s 0. LEMMA 7. Let F [ Ž F1 , . . . , Fn . g ŽCw X 1 , . . . , X n x. n detŽ J Ž F .. g CU . Then

such that

Ži. F1 , . . . , Fn4 contains n y 1 algebraically independent polynomials o¨ er the field CŽ X i . for 1 F i F n. Hence we can uniquely Ž again up to a factor in CU . choose the minimal polynomial M Ž X i , Y . g Cw X i , Y x of F1 , . . . , Fn o¨ er the field CŽ X i . as an irreducible polynomial in Cw X i , Y1 , . . . , Yn x. Žii. For each x i g C, F1Ž X i s x i ., . . . , FnŽ X i s x i .4 contains Ž n y 1. algebraically independent polynomials o¨ er C.

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Žiii. For all but finitely many x i g C, M Ž x i , Y . is the minimal polynomial of F1Ž X i s x i ., . . . , FnŽ X i s x i . o¨ er C. Živ. For x i g C such that M Ž x i , Y . is reducible in Cw Y x, M Ž x i , Y . is a multiple of the minimal polynomial of F1Ž X i s x i ., . . . , FnŽ X i s x i . o¨ er C in Cw Y x. Proof. Ži. The transcendence degree of CŽ F . over CŽ X i . is 1 less than the transcendence degree of CŽ F . over C, which is n. Žii. Set F j [ Ž F1 Ž X i s x i . , . . . , Fjy1 Ž X i s x i . , Fjq1 Ž X i s x i . , . . . , Fn Ž X i s x i . . for 1 F j F n. Since detŽ J Ž F .. g CU , expansion of the determinant according to the partials of F with respect to X i shows that detŽ J Ž F j .. / 0 for some j. Žiii. and Živ. By Lemma 6. Notation. From now on we denote the minimal polynomial Žup to a factor in CU . from the last paragraph by Mi Ž X i , Y . s Mi Ž X i , Y1 , . . . , Yn .. LEMMA 8. Let F [ Ž F1 , . . . , Fn . g ŽCw X x. n with detŽ J Ž F .. g CU and let Mi Ž X i , Y . be the minimal polynomial of F1 , . . . , Fn o¨ er CŽ X i .. Then Mi Ž uX q ¨ , Y . is the minimal polynomial of F1X , . . . , FnX o¨ er CŽ X i ., where u g CU , ¨ g C, FjX Ž X . s Fj Ž X 1 , . . . , X iy1 , uX i q ¨ , X iq1 , . . . , X n . ,

j s 1, . . . , n

with detŽ J Ž F X .. g CU Ž here F X s Ž F1X , . . . , FnX ... Proof. Straightforward. Let F g ŽCw X x. n. Then

LEMMA 9.

Ži. F is birational Ž that is, CŽ F . s CŽ X .. with Fy1 Ž Y . s

ž

U1 V1

,...,

Un Vn

/

g Ž CŽ Y . .

n

Ž Ui , Vi g C w Y x , gcd Ž Ui , Vi . s 1 . if and only if for 1 F i F n, Mi Ž X i , Y1 , . . . , Yn . s a i Ž Vi Ž Y . X i y Ui Ž Y . .

Ž Ui , Vi g C w Y x , gcd Ž Ui , Vi . s 1 . , U

where a i g C .

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Žii. F is an automorphism with Fy1 Ž Y . s Ž G1 , . . . , Gn . g Ž C w Y x .

n

if and only if Mi Ž X i , Y1 , . . . , Yn . s a i Ž X i y Gi Ž Y1 , . . . , Yn . . , where a i g CU . Proof. See, for instance, Theorem 5 in w11x. PROPOSITION 10. Let f 1 , . . . , f n4 ; Cw X 1 , . . . , X ny1 x contain Ž n y 1. algebraically independent polynomials o¨ er C and let GŽ Y1 , . . . , Yn . g Cw Y1 , . . . , Yn x be the minimal polynomial of f 1 , . . . , f n o¨ er C. Suppose V [ Ž c1 , . . . , c n . g C n < f 1 s c1 , . . . , f n s c n ha¨ e a common solution g C ny 1 4 and V Ž G . [ Ž c1 , . . . , c n . g C n < G Ž c1 , . . . , c n . s 0 4 . Then V Ž G . is the Zariski closure of V. Proof. By Lemma 4, GŽ Y1 , . . . , Yn . is the generator of the principal ideal

Ž ² f 1 y Y1 , . . . , f n y Yn :C w X1 , . . . , X ny1 , Y x . l C w Y x of Cw Y x. By Theorem 2.5.3 in w1x, V Ž G . is the Zariski closure of V. PROPOSITION 11. Let F g ŽCw X x. n with detŽ J Ž F .. g CU and let wCŽ X . : CŽ F .x s m. Then for a nonempty Zariski open set of points c g C n, V Ž F y c . [ x g C n < F Ž x . s c4 contains exactly m points in C n, and for all c g C n it contains at most m points. Proof. See, for instance, Lemma 3 in w4x.

3. PROOF OF THEOREM 1 AND THEOREM 2 Proof of Theorem 1. By Lemma 6, for all but finitely many x i g C, Mi Ž x i , Y1 , . . . , Yn . is the minimal polynomial m i Ž x i , Y . of F1 Ž X 1 , . . . , X iy1 , x i , X iq1 , . . . , X n . , . . . , Fn Ž X 1 , . . . , X iy1 , x i , X iq1 , . . . , X n .

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over C. Suppose on the contrary, m i0 Ž Y ., . . . , m i d iŽ Y . have a common zero c g C n. Hence for all but finitely many x i g C, m i Ž x i , c . s 0. Therefore the system of equations F Ž X 1 , . . . , X iy1 , x i , X iq1 , . . . , X n . s c has a zero in C ny 1 for infinitely many x i g C by Proposition 10. Namely, V Ž F y c . contains an infinite number of points, a contradiction to Proposition 11. Proof of Theorem 2. For any fixed i, define a ring homomorphism f : Cw X i , Y x ª Cw X x naturally induced by f Ž X j . s Fj for all j / i and f Ž X i . s X i . Obviously Mi Ž X i , Y . generates the kernel of f. By Theorem 1 in Nagata w6x, Theorem 1 implies that Cw Y, X i xr² Mi Ž X i , Y .: is flat over Cw Y x. Applying f, Cw F, X i x is flat over Cw F x. Remark. When n s 2, Theorem 1 and Theorem 2 become Theorem 5 and Theorem 6 in Formanek w4x, respectively. The proofs presented here are motivated by Formanek’s idea.

4. PROOF OF THEOREM 3 AND THEOREM 4 LEMMA 12. Let F g ŽCw X x. n : C n ª C n be a polynomial map. Then F is birational with detŽ J Ž F .. g CU if and only if F is injecti¨ e. Proof. Suppose F is birational with detŽ J Ž F .. g CU . By Proposition 11, F is injective. Conversely, if F is injective, by Fact 2 in w7x, detŽ J Ž F .. g CU . By Proposition 11 again, F is birational. Proof of Theorem 3. By Lemma 9, for a fixed i, 1 F i F n, Mi Ž X i , Y . s m i0 Ž Y . q m i1Ž Y . X i . Suppose m i1Ž Y . f CU . By Lemma 7 and Lemma 6, if necessary we may replace F Ž X 1 , . . . , X n . by F Ž X 1 , . . . , X iy1 , X i q c, X iq1 , . . . , X n . with some appropriate c g CU ; hence replace m i0 Ž Y . by m i0 Ž Y . q cm i1Ž Y . and this new m i0 Ž Y . is the minimal polynomial of F1Ž X i s 0., . . . , FnŽ X i s 0.. Hence without loss of generality, we may assume degŽ m i1Ž Y .. F degŽ m i0 Ž Y .. and m i0 Ž Y . is the minimal polynomial of F1Ž X i s 0., . . . , FnŽ X i s 0. over C, where deg stands for the total degree. By Theorem 1, m i0 Ž Y . and m i1Ž Y . have no common zeros in C n. Since m i0 Ž F Ž X i s 0.. is identically zero, it follows that m i1Ž F Ž X i s 0.. can have no zeros, hence m i1Ž F Ž X i s 0.. s e g CU . Since m i0 Ž Y . is the minimal polynomial of F1Ž X i s 0., . . . , FnŽ X i s 0., we get m i0 Ž Y .< m i1Ž Y . y e. Since degŽ m i1Ž Y .. F degŽ m i0 Ž Y .., we get m i1Ž Y . s bm i0 Ž Y . q e, b g CU . If b

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s 0, then m i1Ž Y . g CU , contradiction. If b / 0, replace F Ž X . by F X 1 , . . . , X iy1 , X i y

ž

1 b

/

, X iq1 , . . . , X n .

We get m i0 Ž Y . s yerb g CU , which is impossible since by Lemma 7, m i0 Ž Y . is divisible by the minimal polynomial of F1Ž X i s 0., . . . , FnŽ X i s 0. over C, contradiction. Hence m i1Ž Y . g CU for any fixed i, 1 F i F n. By Lemma 9 again, F is an automorphism. Proof of Theorem 4. By Theorem 3 and Lemma 12. Remark. By Lemma 12, Rudin’s paper w7x also yields another simple proof of Theorem 1.

ACKNOWLEDGMENTS The author would like to thank Gary Meisters for bringing his attention to the topics by sending a copy of Walter Rudin’s recent paper w7x in September 1995. He also thanks Edward Formanek for kindly handing him a preprint of w3x in June 1993. Finally, the author is grateful to L. Andrew Campbell for carefully reading and revising the manuscript and for stimulating discussions.

REFERENCES 1. W. Adams and P. Loustaunau, ‘‘An Introduction to Groebner Bases,’’ Graduate Studies in Mathematics, Vol. 3, American Mathematical Society, Providence, RI, 1995. 2. A. Bialynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 Ž1962., 200]203. 3. A. Bass, E. Connell, and D. Wright, The Jacobian Conjecture: Reduction on degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S. ) 7 Ž1982., 287]330. 4. E. Formanek, Observations about the Jacobian conjecture, Houston J. Math. 20, No. 3 Ž1994., 369]380. 5. O. Keller, Ganze Cremona-Transformationen, Monats. Math. Physik 47 Ž1939., 299]306. 6. M. Nagata, Flatness of an extension of a commutative ring, J. Math. Kyoto Uni¨ . 9 Ž1969., 439]448. 7. W. Rudin, Injective polynomial maps are automorphisms, Amer. Math. Monthly 102, No. 6 Ž1995., 540]543. 8. S. Schinzel, ‘‘Selected Topics on Polynomials,’’ University Michigan Press, Ann Arbor, 1982. 9. W. Li and J.-T. Yu, Computing minimal polynomials and degree of unfaithfulness, Comm. Algebra 21 No. 10 Ž1993., 3557]3569. 10. J.-T. Yu, Face polynomials and inversion formula, J. Pure. Appl. Algebra 78 Ž1992., 213]219. 11. J.-T. Yu, On relations between Jacobians and minimal polynomials, Linear Algebra Appl. 221 Ž1995., 19]29.