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INJECTIVITY OF POLYNOMIAL LOCAL HOMEOMORPHISMS OF R” -f

ANNA CIMA,t. ARMENGOL GASULLP and FRANCESC MAfiOSAS§ $ Departament de Matematica Aplicada II, E.T.S. d’Enginyers Industrials de Terrassa, Universitat Politknica de

Catalunya, Colom 11, 08222 Terrasa, Barcelona, Spain; and 0 Department de Matemhiques, Edifici C, Universitat Autbnoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

(Received 20 May 1994; received,for publication 21 October 1994)

Ke.y words and phrases: Real Jacobian conjecture, injectivity, polynomial map, index.

1. INTRODUCTION

Let F: R” + R” be a differentiable map and denote by JF(x) its Jacobian matrix at the point x E R”. It is well known that if det(JF(x)) f 0 for all x E R” then the map F does not need be invertible, although it is a local diffeomorphism at every x. We say that a map F is polynomial if its n components are polynomial functions. This paper mainly deal with the problem to ensure when a polynomial map with nonvanishing Jacobian is a global diffeomorphism.

The results that we get are coherent with the following two famous conjectures.

Real Jacobian conjecture [l-3]. Let F: R” -+ R” be a polynomial map such that the determinant of its Jacobian matrix never vanishes. Then F is a diffeomorphism of R” onto R”.

Jacobian conjecture [4]. Let F: C” + C” be a polynomial map such that the determinant of its Jacobian matrix is a constant different from zero. Then F is a diffeomorphism of C” onto C” and F-’ is also a polynomial map.

It is well known that to prove both conjectures it suffices to show the injectivity of F (see [I, 5,6]). Furthermore, from this last assertion it is not difficult to prove that if the Real

Jacobian conjecture holds then the Jacobian conjecture also holds: let F: C” -+ C” satisfying the hypotheses of the Jacobian conjecture and let F*: R2” -+ R2” be the map obtained from F by identifying C” with R2”. It is easy to comprove that det(JF*(x)) = (det JF(z)(‘, where x and z are the corresponding points of R2” and C” by the above identification. Therefore, the map F* verifies the hypotheses of the Real Jacobian conjecture. So if the Real Jacobian conjecture holds, then F* is one to one and, hence, F is one to one. Then the implication is proved. From now on, we will concentrate on the Real Jacobian conjecture.

In [7, 81 the authors give sufficient conditions to ensure that a map satisfying the hypotheses of the Real Jacobian conjecture will be one to one. In the present paper we obtain more general conditions and also we prove injectivity with slightly different conditions. To state the results we need introduce some notation and definitions.

TSupported by a DCKIC‘YT grand number PB93-0860.

877

878 A.C‘IMAer al

Set m = (m,, ml, . . ., m,), where m, are positive integers for i = 1,2, . . . , n. We denote by x,,, the set of all polynomial maps F = (P’, . . ., P”): R” 4 R” such that deg P’ I mi for i= 1 , .-., n. By identifying xm with R”, where M is the number of all coefficients of P’, . . . , P” we endow xm with the so-called coefficient topology. Let 3, be the set of all polynomial maps F E xm such that the determinant of the Jacobian matrix JF(x) is different from zero at each x E R”.

Let P E R[x, , . . . , x,] be a polynomial, and set r, w,, . .., w, positive integers. We will say that P is quasihomogeneous of quasidegree r with weight w = (wl, . . . , w,) if P(A”lx, , . ., lWnxn) = I’P(x,, . . . , x,) for all (x, , _. ., x,) E R”. Notice that if wr = ... = w, = 1 then the above definition is the usual definition of homogeneous polynomial of degree r. Clearly, if we fix a weight w, then every polynomial can be decomposed as a sum of quasihomogeneous polynomials of different quasidegrees, with weight w. We denote by P, the quasihomogeneous component, with weight w, of maximum quasidegree of P. If F = (P’, . . . . P”) E x,,* we also denote by F, the map with components (P’ w, .*., PG).

We will say that F E 6, if it exists a weight w, such that 0 is the unique real zero of F,. Note that this condition is equivalent to say that 0 is a real isolated zero of F,.

Our main results are the following theorems.

THEOREM A. If F E 3, n G,,, then F is one to one.

Notice that theorem A is a generalisation of theorem A of [7], and of results of [S] because of the more general notion of quasihomogeneous map. The following example shows this difference: consider the map G(x,y, Z) = (x -y’, x + y + y’, z + pk(x, y) + z?+‘), in $c2,3,2k+,), where pk is a polynomial of degree k 1 1. If we take w = (1, 1, 1) we have G,(x, y, z) = (-y2, y3, zZk+’ ) which has nontrivial zeros. However, if we take w’ = (2, 1, l), then we get G,s(x, y, z) = (x - y’, y3, zzk+’ ) which has only (0, 0,O) as a real solution. So our theorem also works in this case and neither the results of [7] nor the results of [8] imply injectivity of G.

On the other hand, we point out that there are one to one polynomials F E $,,\$j,. For instance if we take F(x,,v) = (y + y3, x + xv’), then F satisfies the hypotheses of the real Jacobian conjecture, is one to one but F $ s(1,3) (for any weight w = (w,, w,), F, always is (y’, xy2) which has the zero nonisolated).

THEOREM B. The following statements hold: (1) if F E int(&) then F is one to one; (2) Int(&) # @ if and only if m, , . .., m,, are odd numbers.

Observe that the above theorem implies the injectivity for “most” polynomial maps of 3,. The proof of the above result uses a characterization of the Int($,) (see lemma 4)

that can be used to obtain examples of one to one maps in 3,. Take for instance F,(x, y) = (-x - x3 - xy* + &p3(x,y), -y + x3 - x”y ~ y3 + &q3(x,y)), where p3 and q3 are poly- nomials of degree 3. From lemma 4, FO E lnt(g (3, 3,). Hence, for E small enough F, E $(3,3) and from theorem B it is one to one.

The following is a new result concerning global injectivity for quasihomogeneous polynomial maps. This result generalizes the following easy assertion: if F: R -+ R is a homogeneous polynomial map then F is a homeomorphism if and only if det(JF(x)) = F’(x) does not change sign and vanishes only at 0.

Polynomial local homeomorphisms 879

THEOREM C. Let F = (P’, . . . , P”): R” --) R”. Assume that for all i E (1, . . . , n], Pi is a quasihomogeneous polynomial map with weight w.

Assuming n > 2 the following assertions hold: (1) if det(JF(x)) # 0 for all x # 0 then F is a homeomorphism; (2) if F is one to one then det(JF(x)) does not change sign. Furthermore, there are maps with

det(JF(x)) without changing sign and not one to one and maps satisfying the same property and one to one.

Assuming n = 2 the following assertions hold: (1) if det (JF(x)) # 0 for all x # 0 then Cardjy : F(y) = al is constant for all a # 0.

Furthermore, Card (y : F(y) = 0) = 1; (2) given a positive integer number k there is a map F with det(JF(x)) # 0 for all x # 0 such

that Card(y : F(y) = a) = k for all II # 0.

The proof of (2) for the case n > 2 uses the following result for (3’ maps which we think is interesting by itself.

THEOREM D. Assume that F: R” -, R” is a C’ map such that at some point a, det (JF(a)) = 0 and at any neighbourhood of a, det (JF(x)) takes both positive and negative values. Then, at this point, F is not locally one to one.

2.PROOFOFTHETHEOREMS

Let q E R” be an isolated zero of F. Let B, be a closed ball centered at 4 such that F(x) # 0 for all x E B,, x # q. Clearly, the boundary of B4 can be identified with Sn-‘, the (n - 1)-sphere. Thus we can consider the map F,: S”-’ 4 Sn-‘, defined by F,(x) = F(x)/ 1 F(x) I. The index of F at q is defined as the homological degree of F,. It is well known that if det JF(q) # 0 then the index of Fat q is equal to I (resp. - 1) if det JF(q) is greater (resp. less) than zero.

We denote by iF(q) the index of Fat q. When F has finitely many real zeros, we denote by C iF the sum of the indices of X at all of them.

The following result is well known (see [9]).

LEMMA 1. Let F and G be continuous maps defined on the closure of a open connected and bounded set U of R”. Assume that F and G are homotopic on the boundary 6U (that is, there is a continuous homotopy H(t, z): [0, l] x i? + R” between F and G, such that H(t, z) # 0 for all z E 6U). Then the sum of the indices of the zeros of F and G inside Cl coincide.

The proof of theorem A is based on the following two propositions.

PROPOSITION 2. Let FE S,, and let w be a weight such that 0 is an isolated solution of F,(x) = 0. Then if F has finitely many real zeros

C i,. = i,.“,(O).

880 A. CIMA et ul

Proof. Set w = (w, , . . ., w,,), and for i = 1, . . ., n set W1 = wr ... w,/wi. Then the set

A = ((x,, .,.,x,) E R” :x;“’ + ... + x;*, I 1)

is a compact subset of R”. We write F = (F’, . . ., F”) and F, = (F’ WY ..-, FJ). Let x = (x1, . . ., x,,) E &l. Then, for some i E (1, . . . , n) we have F;(x) # 0. Since Fb is the quasi- homogeous component of maximum degree of F’ we get that there exists I,, such that for all 1 > A,, IF:(lW1x,, . . . . ,IWnx,)l > I(F’ - Fi)(A”lx,, . . . . iL”nx,,)l. From the continuity of F’ and FL this inequality holds in a small neighbourhood U, of x on &I. The compact set &4 can be covered by finitely many neighbourhoods U,,, i = 1, . . . . k. Set I, = max(L,, . . . . AXa). Clearly, for all x = (x, , . . , x,) E A and for all A > ,I0 there exists i E (1, . . . , n) such that

IFj,(A”‘x, , . . ., AWnx,)l > [(F’ - F;)(Px,, . ..) A”nx,)l.

Hence

F,(;l”‘x, , ., A”“x,) + t(F - F,)(A”‘x, , . . . . Awnx,) # 0 for all t E [0, 11.

Lastly, let A > A, be such that the set

A, = ((l.“‘x,, . ..( lWnxn) :x E A 1 = (x E R” : ,+ + . . + x,z*n 5 AZ”’ -.“n)

contains all the zeros of F. Then we get that F and F, are homotopic on the boundary of A, and from lemma 1, the sum of the indices at the zeros of F and F, inside Ax are the same. Since 0 is the unique zero of F, and A, contains all of zeros of F we obtain the desired result. n

At this point we would like to point out the relationship between our proof and another usual approach to prove one to ones for P’ maps. Remember that Hadamard’s theorem (see [lo, p. 301) asserts that a continuous map F from R” to Rn is a homeomorphism if and only if it is proper (i.e. the preimage of compact sets, are compact sets) and it is a local homeomorphism at every point. Similar arguments to those used in the proof of proposition 2 allow us to ensure that under the same hypotheses IF(x)1 goes to infinity when 1x1 goes to infinity. Therefore, we can conclude that F is proper and so that it is a homeomorphism. Anyway, we continue our proof of theorem A using the approach of index theory because it is self contained and it can also be used to prove theorems C and D. We also note that if F is in the hypothesis of (1) of theorem C with n > 2 it can also be proved that F is proper, but since det(JF(0)) = 0, Hadamard’s theorem cannot be applied directly. In this case we could use the results of Church and Hemingsen (see [l 11) that assert that a (!1’ map F from R” to R” with n > 2 and det(JF(x)) # 0 except for an isolated point x0 in which det JF(x,) = 0 is a local homeomorphism also at this point.

The following result is a generalisation of proposition 2.2 of [7]. See also [12].

PROPOSITION 3. Let X be a topological space simply connected and let F: X --t X be a local homeomorphism such that Card F-‘(b) is finite and does not depend on b E X. Then F is a homeomorphism.

Proof of the theorem A. Let F E S,,, f-13, and let w be a weight such that F,(x) has 0 as an isolated real zero. We take b E R” and we claim that K(b) = Card F-‘(b) does not depend on 6. First of all, we note that this number is finite. If not, from Bezout’s theorem, there are no isolated solutions of the system F(x) = 6. So, these solutions must be degenerated (that is,

Polynomial local homeomorphisms 881

the determinant of the Jacobian at these points vanishes), and this contradicts our hypothesis. Let Fb = F - 6. Then, clearly, F,” = F,, and Fb E s, fl&. Notice that K(b) is equal to the numbers of zeros of Fb. Since Fb E 3, all of its zeros have the same index: either 1 or -1. Then K(b) = / C i& . Then from proposition 2 we get K(b) = IiF,, and the claim is proved. Now the theorem follows from proposition 3. n

To prove theorem B we need the following lemma which can be proved as proposition 3.4 of [7]. Let m = (m,, . . ., m,) and F = (P’, . . . , P”) E xrn. We denote by F” the map with components (PA, , . . , PG,), where PA; denotes the homogeneous part of Pi of degree mi.

LEMMA 4. Let F E d,,, Then F E lnt($,) if and only if det JF”(x) # 0 for all x # 0.

Proof of theorem 9. First we prove (1). To do this we will see that if F E Int(J,) then 0 is the unique real zero of F,, where w = (1, . . . , I) and so F E 6,. Note that in this case F, = F”. Suppose, to arrive a contradiction that Fw(a) = 0 for some a # 0. Then F,(k) = 0 for all A E R. So x is a nonisolated zero of F,(x) = F”(x). Then det JFm(a) = 0, which gives the desired contradiction.

Now we prove (2). First assume that for some i E 11, . . . , n), mi is even. If F E Int(d,) then, clearly, deg(P’) = mi. From (l), F is one to one and hence the number of real solutions of the system F(x) = 0 is one. If all the solutions of this system in the complex projective space are isolated then from Bezout’s Theorem this system has, counting multiplicities, m, x ... x m, solutions, which is an even number. Since complex solutions appear in a pairs this implies the existence of a real infinite solution, i.e. a nontrivial real solution of F”(x) = 0. By the arguments of the proof of (1) we get a point a # 0 with det JF”(a) = 0, which gives a contradiction with lemma 4. Hence, there is some solution of F(x) = 0 which is not isolated. However, this condition is not stable by small perturbations of the coefficients and this gives a contradiction with the fact that FE Int(&,).

Assume that m, is odd for all i. Next, we construct a map F E Int($,). To do this first we note that if for some i E Il. . . . , nl, m, = 1 the problem reduces in one dimension. So we can assume that m, 2 2 for i = 1, . . . . n. Let

G(x, , . . , x,) = (x”” ‘.Y, , . , xfNtl- ‘xn),

where x“ = (x: + ... + x:)‘~‘. We claim that for any x f 0 all the eigenvalues of JG(x) have positive real part. We have

JG(x) =

\

7

I where A, = (m, - 1)~““~‘.

882 A. CIMA el al.

Clearly, JG( 1, 0, . . . , 0) has all eigenvalues with positive real part. Then if for some x E R”, JG(x) has some eigenvalue with negative real part, necessarily there exists a point z E R” with some eigenvalue purely imaginary. To see that this situation is not possible it suffices to show that for all x E R” the matrix (JG(x))* has no negative real eigenvalues. We also note that our problem in the case that xi = 0 for some i E ( 1, _. . , n), reduces to an equivalent problem in less dimension. So in our considerations we always assume that xi # 0 for i = 1, . . . , n. By easy computations, we get that (JG(x))* is

n,(l,a: + ..’ + n,1-;,x: I,(i,x,u, + .” * i,x,Lrw,x, .” A:,(n,x,a, + .” + a,x”a,)x~x,

i,(l,x,~l, + ... + i.,x,L)x,.x2 A&,x; + ‘.’ + a,x;,x; ..’ I,(A,x,2 + ... + ~nx,a,)x”xz

\i,(l.,x,~, + ... + ~,,.Y,~,)x,x, i,(l,xf t ..' c A,x,a,)x2xn ... 4(a,Xf+ ". +&4-C

where (Y; = x*/((m, - 1)x;) + x,.

As is well known, a sufficient condition to ensure that a matrix M = (aij) has no negative real eigenvalues is that all diagonal minor (that is, a minor constructed with the terms aij when (&A E l.i,, . . ..j.l x lj, , . . ..j.) and 1 ~j, <j, < ... < j, 5 ra) have positive determinant. Let B be a diagonal minor of JG(x). Without loss of generality we can assume that B is formed by the first k files and the first k columns of M. Then we can write B as

where K = Ak+,xi+, + .‘. + 1,x,“. Then we get det B = A, . .. l&,x: ... x,f det C. where

/ L,uf - .‘. t a,.$ + K i,x,a, + ... t A,x; + K ..’ i.,x,a, t .‘. f I,x,a, t K

\

C= A,$ + ... +,.,x; + K ... Al,x; + ... + I,x,a, + K

‘..

Notice that C is a symmetric matrix and, hence, for our purposes it suffices to show that C is definite positive as a quadratic form. Since the quadratic form

D=

/ K K .” K

K K ... K

\K K ... K

Polynomial local homeomorphisms 883

is semidefinite positive and C = D + E, where

A,x,a, + ‘.. + n,x,’ ... i,X,cwl + ... + AkX/&Yk

A,xf + ..’ + &Xi ... E=

&x,2 + f-f ,

. .

to see that C is definite positive it is enough to show that E is definite positive. Set

w=

It is easy to see that Wr W = E, where Wr denotes the transpose of W. Therefore, yTEy = yTWTWy = (WY)~W~ 2 0 and y’Ey = 0 if and only if W(y) = 0. So to finish our proof it suffices to show that det W # 0. We have det W = Gx, e.0 xk det W,, where

w, =

aI - 1 . . . 1 XI

1 s . . . 1

x2

: : . . . . . .

1 1 . . . k xk I II

.” 1+&k

and E; = X2/((mi_l)X,2) > 0. To see that det W, # 0 we show by induction that det W, > 0. For k = 1 we have det WI =

1 +~,>O.Assumethatdet~>Ofori= l,...,/.Thenweget

= E, det IV-, + e,e2, . . . . E,~,

which is positive by the induction hypothesis. Thus the quadratic forms E and C are definite positive. Hence the determinant of each

diagonal minor of (JG(x))’ is positive and so JG(x) has all its eigenvalues with positive real part for all x # 0. Now we consider the map F = G + I, where I is the identity map of R”. Clearly, JF(x) has all eigenvalues with positive real part for all x E R” and, hence, det JF(x) > 0 for all x E R”. On the other hand, F”’ = G and det G(x) > 0 for all x # 0, so from lemma 4, FE Int(d,) and Int(&,,) # 0. n

884 A. CIMA rt ul

Proof of theorem C. Assume that n > 2. In order to see (l), first we will prove that Cardly : F(y) = 0) = I and Cardly : F(y) = a) is independent of a, for all a # 0. Let y # 0 be such that F(y) = 0. Since F is quasihomogeneous with weight w then F(A”“y,, . . . , AWnyn) = 0 for all A E R. Therefore, det(JF(y)) = 0, and we have a contradiction. Hence CardlF(y) = 0) = 1. Now set X = R”\lO]. From the above proof we can consider F: X + X which is a local homeomorphism at each point of X. Note that if det(JF(x)) # 0 for all x # 0 then det(JF(x)) does not change sign. Therefore, from proposition 2, we have that for each (I # 0

Card F-‘(a) = \ c iF-,l = IiF

which does not depend on a. Since for n > 2, X is simply connected, using the above results we can apply proposition 3

to F: X + X and conclude that F: X --f X is a global homeomorphism and so F: R” + R” is one to one. At this point we recall the following result (see [lo, p. 3, 131); “If fi is an open set in R” and F: Q + R” is one to one and continuous, then F(Q) is open and F is a homeo- morphism.” Hence F: R” -+ R” is a homeomorphism.

The first part of (2) for n > 2 follows from theorem D. The examples x E R” --t (x:,x:, . . . ,x,‘) (one to one) and (x, z) E R” x C = R”+* -+ (x, z*) E Rn+* (non one to one) finish the proof of the theorem for n > 2.

Assume now that n = 2. Part (1) follows arguing as in the proof of (1) for n > 2. The map F(z) = zk, z E C as a map from R* to R* proves (2). n

The following lemma asserts that in some cases, if F is a map with quasihomogeneous components its Jacobian vanishes at some y # 0.

LEMMA 5. Let F = (P’, P*, _. . , P”): R” + R”, n > 2 be a quasihomogeneous polynomial map with weight w, and let T, be the quasidegree of P’. If n “= , r;/wi is even then det(JF(x)) vanishes at some y f 0.

Proof. Assume that det(JF(x)) # 0 for all x # 0. Then from the proof of theorem C, Cardjy : F(y) = a) = i,(O), when a f 0. From [14], ~~(0) = n y=, ri/‘Wi, where ~~(0) denotes the multiplicity of F at zero, and from [15], iF(0) = ,u~(O), (mod 2). Therefore, Cardly : F(y) = a] is an even number and F can not be a homeomorphism, in contradiction with theorem C. n

Remark 6. For n = 2 there are homogeneous maps with nonvanishing Jacobian for all x # 0 and fl f= , r,/w, = I] f= , deg 6 even. Take for instance F(z) = I*~, z E C, as a map from R2 into R*.

Proof of theorem D. It is not restrictive to suppose that F(a) = 0. Take ]~,l,.~ and (ynJnEN two sequences with limit a and such that det(JF(x,)) > 0, and det(JF(y,)) < 0. Assume that F is locally injective, therefore, F(x,,) and F(y,) are regular values (because at these points F-’ takes only simple preimages in a neighbourhood of a). Recall that for each E, regular value of F and sufficiently small

i&J = c w(det JFW),

where the sum is over the set of all the points p in a neighbourhood of II such that F(p) = E. Hence, the above index must be 1 (taking E = F(x,)) or - 1 (Taking E = F(y,)). Thus we have a contradiction and so F is not locally one to one. n

Polynomial local homeomorphisms 885

Notice that if in a neighbourhood of a point a with vanishing Jacobian, det(JF(x)) does not change sign, F may be locally one to one or not, as the examples used in the proof of theorem C show. On the other hand, if in any neighbourhood of a there is an open set with det(JF(x)) = 0 (including the point a), an application of the Sard’s theorem led us to say that F is not locally one to one at LI.

Acknowledgement-We wish to thank C. Olech for his suggestions.

Note added in proof

During the printing process S. Pinchuk gave an example in RZ of a polynomial map of degree 25 satisfying the hypothesis of the Real Jacobian conjecture which is not one to one. After his example the results of this paper on the Real Jacobian conjecture can be understood as some answers to the question: which additional assumptions are necessary for a polynomial map with nonvanishing Jacobian to be a homeomorphism?

I.

2. 3.

4. 5.

6. 7.

8. 9.

IO. 11. 12. 13. 14.

15.

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