THE DIVERGENCE-FREE JACOBIAN CONJECTURE IN DIMENSION TWO

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1 THE DIVERGENCE-FREE JACOBIAN CONJECTURE IN DIMENSION TWO J. W. NEUBERGER Abtract. A pecial cae, called the divergence-free cae, of the Jacobian Conjecture in dimenion two i proved. Thi note outline an argument for a pecial cae of the Jacobian conjecture in dimenion two: Suppoe F : C 2 C 2 i a polnomial o that F (0) = 0, F (0) = I, det(f (z)) = 1, z C 2. (1) where I i the identit tranformation on C 2.Write r(x, )+x F (x, ) =, (x, ) C 2 (x, )+ where r, have no non-zero contant or linear term and oberve that det F = {r, } + +1 o that (1) give {r, } + = 0 (2) with {r, } = r 1 2 r 2 1, = r 1 + 2, the Poion bracket and divergence repectivel of the vector field (r, ), ubcript in thee intance indicating partial derivative in firt and econd argument. The main purpoe of thi note i to prove the following Theorem 1. Suppoe in addition to the above that {r, } =0, = 0. (3) Then F i bijective, i.e., the Jacobian conjecture hold in thi cae. Date: 18 November Mathematic Subject Claification. 14R15. Ke word and phrae. Jacobian Conjecture, Divergence-Free. 1

2 2 J.W.NEUBERGER We haten to point out that (3) doe not follow (no matter what thi writer ma have upected for ome time) automaticall. The following example communicated b Hman Ba howed the author expectation to be fale. Example 1. Suppoe each of a and b i a poitive integer greater than one and ( (x a + ) b ) + x F (x, ) = x a, (x, ) C 2. + It i ea to check that (1) hold but that (3) doe not. Neverthele the following indicate intance in which (3) hold. Corollar 1. Suppoe that all term of F of degree higher than one are even. Then (3) hold. Corollar 2. Suppoe that m>nare poitive integer and (n 1) 2 >m 1. If F i uch that all of it term of degree higher than one have their degree in [n, m], then(3) hold. In both cae there i no term in {, r} whichhaadegreeincommonwitha term of (r ). Hence thee two quantitie that um to zero mut each be zero. Therefore (2) implie (3) in thee two cae. Reference on the Jacobian Conjecture are [1],[4],[5]. An argument for Theorem 1 i baed on the following. Theorem 2. In order for (3) to hold, it i necear and ufficient that there be a linear tranformation L : C 2 C and a polnomial Q : C C 2 o that F (z) =z + Q(Lz) and L(Q(Lz)) = 0, z C 2. (4) Before a proof of Theorem 2, three lemma are given which indicate that if (4) hold then Theorem 1 follow. Lemma 1. If (4) hold, M C 2 and F (M) i bounded, then M i bounded, i.e., F i proper. Proof. [Lemma 1] Suppoe M i a ubet of C 2 and F (M) i bounded. Since L(F (M)) i then bounded, (4) ield that L(M) i alo bounded. But then Q(L(M)) i bounded and o then i M. Lemma 2. Under (4), uppoev, w C 2. There i a unique function u :[0, ) C 2 uch that u(0) = w, u (t) = (F (u(t))) 1 (F (u(t)), t 0. (5) Moreover, q = lim u(t) exit and F (q) =v. (6) t Proof. [Lemma 2] Suppoe that each of w, v C 2 and the equation in (5) hold with olution u for ome maximal interval [0,c),c>0. Then (F (u) = (F (u) on[0,c)

3 THE DIVERGENCE-FREE JACOBIAN CONJECTURE IN DIMENSION TWO 3 and o F (u(t)) v = e t (F (w), t [0,c). (7) Aume that c i finite. Then F (u([0,c))) i bounded and therefore uing Lemma 1, u([0,c)) i alo bounded. Conequentl, due to (5), u ([0,c))) i bounded too. But thi lat concluion give that p = lim t c u(t) exit. Therefore u ma be extended b continuit to [0,c] and conequentl the olution u can be further extended beond [0,c], contradicting the maximalit of [0,c). Thu c =. It follow that there i a olution u to (5) exactl a tated there. Hence uing (5), u decreae to zero exponentiall and o q = lim u(t) t exit. That F (q) =v follow from (7). For v C 2, define G v : C 2 C 2 o that G v (w) = lim t u(t) whereu atifie (5). Given v C 2,oneathatw i in the domain of attraction of q relative to (5) i.e., (G v ) 1 (q) i thi domain of attraction. Lemma 3. Under (4), F ha an invere defined on all of C 2. Proof. [Lemma 3] Suppoe that v C 2. From Lemma 1,2 it follow that ever member of C 2 i in the domain of attraction of ome z o that of F (z) =v. Denote b S v the preimage of v under F. The collection S v ha no limit point ince uch a limit point would be a place at which F i ingular. Hence from general principle of ordinar differential equation, a domain of attraction of an element of S v i an open et. Now C 2 i not the union of mutuall eparated open et. Hence the domain of attraction of an element of S v i all of C 2 and in fact S v contain jut one point. Thu there can not be two element q, q o that F (q) =F (q) ince two uch element would be ditinct member of S v. Lemma 1,2,3 impl that under (3) F i a bijection and hence Theorem 1 follow from Theorem 2 ince thee lemma follow from Theorem 2. See alo [4] in connection with thi lemma. It remain to prove Theorem 2. Proof. [Theorem 2] Under the hpothee of Theorem 2, {r, } =0. Nowifr, are both zero, the concluion urel hold. Accordingl uppoe that one of r and i not zero, a r. Notethatr i not contant. Denote b (α, β) apointof C 2 at which at leat one of the partial derivative r 1,r 2 i not zero. A claical reult on functional dependence (cf [2] for matter of differentiabilit) and (cf [3], p 426 for functional dependence) give that there i ɛ>0 and an analtic function h with domain the open ball B ɛ (radiu ɛ, center (α, β)) o that not both of r 1,r 2 are zero at an point of B ɛ and Note that then (x, ) = h(r(x, )), (x, ) B ɛ. (8) 2 (x, ) = h (r(x, ))r 2 (x, ), (x, ) B ɛ. (9)

4 4 J.W.NEUBERGER (9) together with (3) ield that r 1 (x, )+h (r(x, ))r 2 (x, ) =0, (x, ) B ɛ. (10) For r atifing the above and (γ,δ) B ɛ denote b ( u function with maximal domain in C o that (u ) r2 u γ = (u,, (0) =. v r 1 v δ Note that r(u, =0 and conequentl, r(u, andh(r(u, ) are contant. Denote the common value of h (r(u, ) b c. Uing (10), u 1 = r 2 (u,. v c Thi implie that direction of member of the range of ( u are contant and hence the range of ( u lie on the (complex) line 1 γ W γ,δ = { + : C}, c δ the line of lope c through ( γ δ). Hence r i contant on the interection of thi line and B ɛ and o b analticit, r i contant on all of W γ,δ. It will be een that each member of the et of line {W γ,δ :(γ,δ) B ɛ } (11) ha lope c. If two of thee line had different lope, the would cro; then ever member of (11) would cro at leat one of thee two and hence r would be contant on all of B ɛ (and hence all of C 2 ), a contradiction. Thu the member of (11) are parallel, all with lope c. Put another wa, r atifie on B ɛ the partial differential equation r 1 + cr 2 =0. (12) Hence there i a function f from a ubet of C to C o that r(x, ) =f(cx ), (x, ) B ɛ. (13) The function f clearl i a polnomial. Hence relation (13) extend b analticit to all of C 2. Moreover, (8), with h now known to be linear and homogeneou (actuall the action of h i jut multiplication b c) mutextendtoallofc 2 and conequentl the relationhip (8) extend to all of C 2.Thetwoextenion noted above give that r(x, ) f(cx Q(x, ) = =, (x, ) C 2, (x, cf(cx where the firt equalit above i b definition. Defining L : C 2 C b x L = cx, (x, ) C 2,

5 THE DIVERGENCE-FREE JACOBIAN CONJECTURE IN DIMENSION TWO 5 one ha P (z) = z + Q(Lz), z C 2. Since LQ(Lz) = 0,z C 2, it i hown that (3) implie (4). Now it i to be hown that (4) implie (3). Chooe Q, L o that (4) hold. Chooe a, b C uch that x x L = ax + b, C 2. Denote b each of g, h a polnomial from C to C o that x g(ax + b) Q =, (x, ) C 2. h(ax + b) If a =0=b, thenr =0= and the concluion hold, o uppoe that at leat one of a, b i not zero. From (4) it follow that ag(ax + b)+bh(ax + b) = 0, (x, ) C 2, and o ag + bh = 0 ince with proper choice for x,, ax + b ma be an member of C. Thu ( )(x, ) = (ag + bh) (ax + b) = 0, (x, ) C 2. Thu (3) hold and the argument i finihed. It ha alread been noted that Theorem 1 follow from Theorem 2. Reference [1] H. Ba, E. Connell, D. Wright, The Jacobian Conjecture; Reduction of Degree and Formal Expanion of the Invere, Bull. Amer. Math. Soc. 7 (1982), [2] T.M. Flett, Differential Anali, Cambridge Univerit Pre, [3] J. Olmtead, Real Variable, Appleton-Centur Math. Ser., [4] W. Rudin Injective Polnomial Map are Automorphim, Amer. Math. Monthl 102 (1995), [5] A. van den Een, Polnomial Automorphim and the Jacobian Conjecture, Semin. Congr. 2, Soc. Math. France (1997), Department of Mathematic, Univerit of North Texa, Denton, TX addre: jwn@unt.edu

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