Conjugation for polynomial mappings

Z angew Math Phys 46 (1995) 0044-2275/95/060872 11 $ 1.50 + 0.20 (ZAMP) �9 1995 Birkh/iuser Verlag, Basel

Conjugation for polynomial mappings

By Bo Deng, and Gary H. Meisters, Dept of Mathematics and Statistics, University of Nebraska, Lincoln, NE 68588-0323, USA (E-mail: [email protected], [email protected]) and Gaetano Zampieri, Dip di Matematica, Universitfi di Messina, Salita Sperone 31, 1-98166 Messina, Italy (E-mail: [email protected])

1. Introduction

In his paper [7] (1939), Keller conjectured the bijectivity of a mapping f: C n ~ C n, with polynomial components, and a constant nonzero Jacobian determinant (see [7], page 301). Moreover, he conjectured that such a function always has a polynomial inverse. These mappings f will be called Keller's functions in the sequel. Without loss of generality we can always assume that f (0) = 0, f ' (0) = I (the Jacobian matrix at the origin is the identity matrix) and

de t f ' (x) = 1, x e C ~. (1.1)

In 1960, Newman [10] in dimension 2, and Biatynicki-Birula and Rosenlicht [3] (1962) in all dimensions, proved that surjectivity and polynomial character of the inverse follow from injectivity.

Another important result in this research field is the reduction o f degree theorem of Yagzhev [13], 1980, also found by Bass-Connell-Wright [2], 1982. This theorem says that it suffices to prove Keller's conjecture for polynomial mappings f: C ~ ~ C ~ of degree d = 3 of the following form:

f ( x ) = x + g(x), g(sx) = sdg(x), g'(x) ~ = O, x ~ C ~, s ~ C (1.2)

(n in g' (x)~= 0 is the same as in C~). The last condition, that is the nilpotence of g'(x) at all x ~ C ~, is equivalent to Keller's hypothesis (1.1) as is well known and easily checked. The functions as in (1.2) for arbitrary d will be called Keller's functions o f the homogeneous form.

The present paper studies some properties of the Keller's functions. We are interested in the existence of certain related functions x ~ h~.(x), where 2 > 1 is a parameter, defined by power series in a ball with center at the origin, such that h'(0) = I and

h)~(2f(x)) = 2h~(x). (1.3)

Vol. 46, 1995 Conjugation for polynomial mappings 873

So each hx conjugates the function ) f t o its linear part )f'(O) = 2I in a ball where it is injective.

For 2 > 1 the local existence of the conjugation hx is guaranteed by the Poincar6-Siegel Theorem (see [1], Section 25, p. 193). Our original goal was to demonstrate that h;, is an entire function. This would imply that ) f is injective and Keller's conjecture by [3]. But we were unable to do so in this paper. Instead, we obtained a proof of the Poincara-Siegel Theorem for the polynomial mappings of constant determinant. Our main results give quan- titative information about the conjugacy function h;. that cannot be derived from the general Poincar&Siegel Theorem.

The main results are stated in Section 2. The proofs are given in Section 3 and 4. In Section 5, we give two examples of Keller's functions f of the homogeneous form (1.2); the conjugation hx is globally defined for any 2 > 1 and it is even a Keller's function. We also consider an example of a Keller's function which leads to conjugations whose local inverses are not polynomial mappings (this example is not of the homogeneous form).

We conjecture that for Keller's functions f of the homogeneous form, the conjugation h~. for 2f is an entire function.

Keller's problem interested many mathematicians of different fields: Algebra, Algebraic Geometry, and Analysis. The literature is vast and we are not going to review it, we address the interested reader to [8] and [9] for discussions and references (the bibliography of [8] has 208 entries). For the general question of the global invertibility of local homeomorphisms see [4], [12], and the references contained therein.

Also the dynamical behavior of Keller's functions interested several authors; here let us just mention the recent paper [6] by Friedland and Milnor.

2. Main results

The main results of this paper are given by the following two theorems.

Theorem 1. Let f: C n ~ C n be a polynomial mapping with constant Jacobian determinant det f ' ( x ) - constant, let f (0) = 0 and f ' (0) = L Let

= max{ k =0 , 1 , . . . } ,

which is finite because all the entries in the matrix f ' ( x ) - 1 are polynomials. Then for each 2 > 16M 2, there is a function hx(x) analytic in ]]x]] < 2/(16M 2) so that the following conjugacy equation

= ( z . 1)

holds in a neighborhood of the origin in which both Ilxll < 2/(16M 2) and II/(x) 11 < 1/(16M 2) are satisfied.

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Theorem 2. The solution h;. of Eq. (2.1) f rom Theorem 1 satisfies the following properties.

(a) h A ( x ) is injective in Ilxll < )~/(64M2). (b) h A ( x ) ~ x as 2 ~ ~ uniformly in any bounded ball centered at the

origin. (c) The inverse h~ -1 of hA is an entire function. (d) The determinant deth~(x) is constant for Ilxll < ) o / ( 1 6 M 2 ) , and

d e t ( h ; s l ( x ) ) ' is constant for all x 6 C ~.

3. Notations

Denote by C ~ • n x . . . x C ~ the k product of C n. Let g: C " - - ~ C ~

be any mapping, denote by g(k) the kth-derivative of g. For each x e C n.

g ( k ) ( x ) : C k x n___+ C n, ( V l , . . . , Vk ) ~_~ g ( k ) ( x ) [ 1 ) l , . . . ' Vk],

is multi l inear in vi. Sometimes we use the following notat ions

g(0) = g, g, = g (1 ) , g , , = g(2), g ( k ) ( x ) v k = g(k)(X)[V, . . . , V],

for v e C ~. Let Iz[ be the absolute value of z e C. Let IIxh[ = max{Ixi]: 1 < i < n } be

the sup no rm of x = ( X l , . . . , x , ) e C n. The operator no rm IIrll for any multi l inear t ransformat ion T: C k • C ~ is defined by

[[T[[ = sup{ l iT[v1 , . . . , vklll= I/v, I[--- 1, i : 1 , . . . , k } .

Thus, we have

l i T [ v , , . . . , vklH < HTH I1~111 I1~ J["

We will also use the following nota t ion for simplicity:

T [ [ u l , . . . , uk] + Iv1 . . . . , vkl] .'= T [ U l , . . . , uk] + T i r e , . . . , vkl .

We will refer to [ u ~ , . . . , uk] as a k-bracket form.

4. Proof of the main results

Let f satisfy the condit ions of Theorem 1 in the sequel. Let

q (x) =if(x)- 1, x e C" .

As ment ioned in the introduct ion, we assume without loss of generality that

f (0 ) = 0, i f (0) = I. (4.1)


Also, because de t f ' (x ) -- 1, x ~ C n, 4~(x) has polynomial components. Thus, there exists a constant M > 1 so that

II (k)(0)]l _< M, for all k = 0 , 1 , 2 , . . . . (4.2)

We will also use the following notation

q~ (x) = { ~b(x), 4 '(x) . . . . , q~ (k)(x) }, (4.3)

the collection of the first k derivatives of qS. The proof of Theorem 1 is divided into several lemmas.

Lemma 3. Suppose for 2 > 1, ha is analytic at x = 0 and satisfies the conjugacy equation (2.1) in a neighborhood of x = 0. Then

h ( ~ ) ( 0 ) - 2 ~ - ' 1 ~ h~k-~ i(e,(0)) , k = 2 , 3 . . . . . - - i = 1

Here Ak,i is the sum of Ck,i mappings called akjj. Each ak,i,j maps k-bracket forms to /-bracket forms, ak,i,j is k homogeneous in the elements of the collection ask ~(0), possibly repeated and

lib? (0)a~,~d(q)k-,(0))I1 < IIhi~ (0) ]l Mk, j = 1, 2 . . . . . Ck,i,

where M is as in (4.2) and ~i(x) is as in (4.3). Moreover the sequence {Ck,~ }, 0 _< i _< k, k = 1, 2 , . . . , satisfy the fol-

lowing recursive formula

Ck+ ~,~ = kG, i + G,~- 1, G,k = 1, G,0 = 0.

Proof. We omit the subindex of h:~ in the proof. We first prove the following identity by induction.

k - 1

) ~ ' h ( k ) ( 2 f ( x ) ) = ~, h (k ~ k = 1, 2 , . . . , (4.4) i=0

where A~ok(q)o(X)) = [q~(x), . . . , qS(x)] = qS(x) k and A k , ~ ( O i ( x ) ) , i = 1 , . . . ,

k - 1 is the same as in the statement of the lemma. For k = 1, differentiate Eq. (2.1) directly and cancel 2 from both sides, we have

h ' ( 2 f ( x ) ) [ f ' ( x ) u ] = h ' (x ) [u] , for all u ~ C".

Replacing u by c~(x)u, we have

h ' ( 2 f ( x ) ) [ u ] = h ' (x)[qb(x)u] ,

which is the case for k = 1. As induction hypothesis, assume (4.4) holds for k. For the case of k + l, differentiating the left side of (4.4), we have

(2 k- ~h (k)(2f(x)) '[U~, . . . , U~, U~ +1 ]

= ( ,~k - lh (k ) ( ; c f ( x ) ) [Ul , �9 �9 �9 , Uk])'[Uk+ 1]

= 2kh (~ + 1 ) ( 2 f ( x ) ) [ U l , . . . , Uk, f ' ( x ) u k +1 ],

876 B. Deng et al. Z A M P

for all ui e C n, 1 < i _< k + 1. Different ia t ing the r ight side o f (4.4), we have

k - 1

~, (h(k-i)(x)Ak,~ i(@i(x))[ul, . . . , ukl)'[u~+l] i = 0

k - 1

= ~ (h(k-i+ 1)(x)[Ak,k _ i(@i (X))[Ul, �9 �9 �9 , U~], Uk +1] i = 0

+ h~ k- O(x)(A~.~_ i(~, ( x ) ) [U l , . . . , u~])'[u~ +11)

= h ~+ 1)[6(x)u1, . . . , ~(x)uk, u~+ 1] k - 1

+ ~ h(k+l-O(X)[[Ak,k_i (* i (X))[Ul , . . . , Uk], Uk+l] i = 1

+ (Ak,k-i+ l(@,-I(X))) '[Ul, �9 �9 �9 Uk, Uk+ 11]

+ h'(x)(Ak, l ((I)k(x)))'[Ul, �9 �9 �9 uk, u~ +1].

The last ident i ty is ob ta ined by collect ing the like te rms o f h ~~ Replace uk+~ above by O(x)uk+l and equa te the two sides for the dif ferent ia ted (4.4), we

have

k

2kh (k+ l)()~f(x)) = ~ h (k+~ -i)(x)Ak+ l,k+, _i(q~i(x)), i = 0

where

A~+ 1,k+1 (~0(x))[u , , . . �9 , uk+ 1] = [ r . . . , O(x)uk+ ,],

Ak+ ,,k+l i ( r . . . , uk+ 1] = [[Ak,k-i(qai(x))[Ul, . . . , u~], O(x)uk+ ,]

+[ (Ak ,k_~+l (@i- l (X) ) ) ' [u , , . . . , uk, qS(x)u~+ ~]], 1 _< i < k - 1,

Ak + 1,1 = (A~,I (aPk(X)))'[Ul, �9 �9 �9 Uk, ~p(X)U~ +1 ].

Because Ak,k_~(r maps k - b r a c k e t fo rms to (k - / ) -b racke t forms, the

a u g m e n t e d b racke t [Ak,k_i(OP~(x))[u~,.. . ,uk], qS(x)uk+l] maps ( k + l ) - b racke t fo rms to ( k + l - / ) - b r a c k e t forms. Because Ak,~_~+~(q~_~(x)) maps k - b r a c k e t forms to ( k - i + 1)-bracket forms, its der ivat ive maps (k + 1)-bracket fo rms to (k + 1 - / ) - b r a c k e t forms. Hence , we conc lude tha t A~+l ,~+1_~(~(x) ) maps (k + D-b racke t fo rms to (k + 1 - / ) - b r a c k e t forms. Mor e ove r , it is easy to see tha t the n u m b e r o f such b racke t - to - b racke t m a p p i n g in each A~+~,~+~_/(r is given by the fol lowing

formula :

C~+~,~+~ = C~,~ = 1,

Ck+l,k+l i=Ck,k_i-~-kCk,k_i+l, l < _ i < _ k - 1 ,

C,+ 1,1 =kC~,~.


The coefficient k for Ck,a-i+l is due to the fact that each of the Ck,k-i+~ mappings a~,k i+ 1j is k homogeneous in the elements of ~i_ l(x) and each derivative gives rise to k bracket- to-bracket mappings. When evaluated at [ul , . . . , u~, 4(x)uk+ 1], each of the k mappings becomes k + 1 homogeneous in ~ ( x ) . This proves identify (4.4). By induction, we also have

[Ih<k +i -o(0)ak + 1,~, +l ,,j(q)i(0))I[ < II h(k + 1-o(0)IIM k + l

Now evaluate (4.4) at x = 0 and use the fact that 4 ) ( 0 ) = / , we can express h(k)(0) as follows

h ' (0 ) = h ' (0) l k - 1

h(k)(0) - 2 k- I 1 ~ h(k-~ k = 2 , 3 , . . . . - - i=1

This completes the proof.

For definiteness, we fix f rom now on

h'(0) = Z.

Lemma 4. Let the sequence { C~,i } of integers be defined as in L e m m a 3. Then

i! k~.C~,~<2k, O<_i<k, k = l , 2 , . . . .

Proof. Let Dk,~,=(i!/k!)C~,~. We prove the estimate by induction. For k = 1, i = 0, 1, DI~,~ = 0 or 1. The estimate is true. Suppose it holds for k and 0 < i < k. Then by the recursive formula for Cz:,~ f rom Lemma 3, we have

D~+ ~,i - (k + 1 ) ~ . t G + 1,i - (k + 1)! (kG, , + G / ~)

k i - k + 1 D~,i + ~ Dk,~_ 1

k + i 2k 2 k 2 k+l < ~ - - ~ < 2 " = , 0 < i _ k + l . []

Lemma 5. Let hx(x) be the power series defined as follows

1 h(x) = x + kZ= 2

where the multi l inear maps h(k)(0) are defined by the same recursive formula f rom L e m m a 3. Suppose

2 > 32M z.

878 B. Deng et al.

Then for I[xtl < r < 2/(16M2),

[Ih~(x) - x]l -< 1 - Qr' 0 -- 16M2/2,

where M is the bound for [l~<~>(0)]], k = 0 , 1, 2 . . . as in ( 4 . 2 ) .

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Proo f of Theorem 1. We drop the subindex in h: in the proof. Let

h(x ) = x + ~. h(~)(O)x ~ k = 2

Proof. F o r simplicity, we drop the subindex f rom hx in the proof. We first prove by induct ion that

1 r[h~k>(0)l I < ~ k - 1 • = 1, 2 , . . . k! - '

Fo r k = 1, we have h ' ( 0 ) = I and ]]h'(0)]] = I < Q ~ Assume the estimates hold for i = 1, 2 , . . . , k. Consider the case for k + 1. By L e m m a 3, we know that Ak + 1,~ + 1 - i contains Ck + 1,k + 1 - i n u m b e r o f ak + 1,k + 1 = i,j mappings and Ilh(~+,-,>(0)a~+l,,~+l ,,ill-< Ilh(~+l o(O)[]Mk+ 1 Thus,

]lh<~ +i-i>( O)Ak + 1,k +I i( ~ i ( 0)) I] < Ck + I,~ +1- i M k + I llh<~ +i-o(o)II. Use this est imate together with L e m m a 4, we have

I i k I (k + I)! ]lh(~+ ~ < ) k ~ i~i (k + I)! Ilh<'+1-i>Ak+"~+1-i(O;(0))l[

1 & ( k + m - i ) ! IIh<'+l-O(0)ll < 2 k - l i : l L ( k + l ) ) . C k + x ' k + l - i M k + l ( k + l - - i ) !

1 < 2 k - 1 ~ 2k+ lMk+10k- i

i = 1

( 2 M ) k+l 1

<- ~ - - - 1 1--0"

Since 0 < e < 1/2, 2 k - 1 > 2k/2, we have

1 ( 2 M ) k+l 1 4 (2M) k+l (16M2) k Qk.

- - < 2k < 2k -

N o w for Ilxll-< r < 1/~ --~/(16M2), we have the desired est imate

1 iih<~>(o)[ I ilxll ~ k = 2

<_ r ~, (or) k i = 0 r2 [] ~=2 1 -- or"


with the multilinear map h(k)(0) defined by the recursive formula of h(k)(0) in Lemma 3. By Lemma 5, h(x) converges in IIx[[ <Rx with R~= 1/O = 2/(16M2).

Since h(x) is analytic in Ilxll < R~, h(2f(x)) is well defined and analytic at all points x satisfying I]f(x)ll < 1/(16M 2) and []xll < R ; . By the proof of Lemma 3,

( h ( ; f ( x ) ) ) ~)lx ~ o = ,~h ~ ' ( o)

for all k = 0 , 1, 2 , . . . . Therefore, the conjugacy equation (2.1) is satisfied for x in a neighborhood of the origin so that I]xll <R~, ]If(x)] i < 1/(16M2). []

Proof of Theorem 2. (a) h(x) is injective in I[xll < R~/4 because of the following argument. Let

g(x) = h ( x ) - x = ~ l h ( k ) ( 0 ) x k k=2

and IFxll, Irylr _<r <R~/4, we have

fo ~ dt I IIg(x) - g(Y)[I = g ' ( tx + (1 - t)y)(x - Y)

fo' [Ih~)(~ +(1-0Y)~ lit dt/Ix-ylr Y (k -1)v 1 <

k = 2

_< kQ k- I f ' k - 1 dt Ilx - y ]l k=2

ds k s= or

0r(2 -- 0r) ~i - - ~ ; ~ ]Ix - y l l

Therefore, for x r y, [Ixl[, rlYll- r < R~/4 -- 1/(40), we have

]]h(x) - h(y)]l = [Ix - y + g(x) - g(y)[I >- [I x - Y [[ - I]g(x) - g(y)]]

(1 Q-r-(2 ---- 0r)'] 1 - 4 Q r >__ ( 1 - 0 r ) 2 } I I x - y [ [ > ( I ~ - ~ . ) 2 H x - y l l >0 ,

implying the injectivity of h in I]x]] < r < R x / 4 = 1/(40). (b) The conclusion follows from the following estimate

0r 2 l ib(x)- xl[ _< l - 0 r ' P = 16M2/2

880 B. Deng et al. ZAMP

from Lemma 5. That is, h(x) ---+ x uniformly in any fixed ball of radius r as ~. ---+ OO.

(c) The conjugacy equation (2.1) is equivalent to

2f(h - ' ( x ) ) = h -1(2x). (4.6)

By Theorem 1 and part (a) above, h -~ exists in a ball B(ro) of radius, say ro, centered at the origin and Eq. (4.6) is satisfied. Because f is an entire function, Eq. (4.6) implies that h ~ can be extended analytically to the ball B(2ro) of radius 2ro. Since 2 > 1, this argument can be repeated indefinitely so that h - 1 is analytically extended to the entire space C n.

(d) Differentiate (4.6) and cancel 2 from both sides, we have

f ' ( h - l (x) )h - l ' (x) = h -1'(2x), det h -I'(x) = det h -1'(2x),

since d e t f ' - 1. Therefore, for each given x, the last identity implies

det h - l ' ( x ) = det h-1'(2 2 )

: d e t h - " ( ~ ) . . . .

4_x = d e t h \2k] ~ det h-l '(0), as k ~ o o .

Because h'(x) = (h-l ' (h(x))) -~, det h'(x) = 1/det h-l ' (0) follows. []

5. Examples

" All Keller's maps of homogeneous form (1.2) we checked give conjugations which ~.re entire functions.

The following example f: C s ~ C 5 was found by Rusek [11] for other purposes

f ( x ) = ( X l , ) ~ 2 , X3 - - X1X2X4 - - X21X5, X 4 + X 1 X 2 N 3 - - X 2 X 5 ' X5

1 - - -- X 2 ( X 3 -JU "~4))"

2

For each 2 a C, #1 , here is &: C 5 ~ C 5, x ~ h ~ . ( x ) = ( & . ( x ) l , . . . , h~(x)5) which satisfies (1.3) on the whole C n

&(X)l = Xl,

&(x)2 = x2,


1 [ ~2 24 h2(x)3 - ( 2 2 - 1)2(22-~ - l ) _ (1 - - - t - ) o 6 ) x 3 - / - ( 2 4 - ])XlX2X 4

- ~ - ( 2 4 - 1)x2x5-~-22x~x2x5~-~x2x2(x4-x3)1,

1)2( 22-]-1 1) [ 22--24 h,~(x)4 = (22 (1 - -~-- 26)x4 -~ - -~-(1 -24)XLX2X3

.2 ] - - A 2 2: x __ X4 ) + ( 2 4 - 1)x~x5 - 22x~x2xs + - 2 x ~ x 2 t 3

l I 2 4 - 1 X2(X 3 -~ X4 ) h2(x)5 = ( )2 _ 1)2(22 @. 1) (1 -- }2 _ 24 + 26)x5 - ~ - T

-~- 2 Xl X3(X4 -- 23) -~- 22X2X2X5 .

This function is bijective and polynomial as well as its inverse. Next, we consider the planar map

f(Xl, X2) = (X 1 ~- (X 2 ~- X12) 2, X 2 ~- X2). ( 5 . 1 )

Equation (4.6), namely 2f(h -1(x)) = h ~(2x), leads to not so trivial calcula- tions for the derivatives of h 1 at 0. They show that h ~ is not a polynomial function.

By reduction of degree (5.1) gives the bijection

~(x) = (x, + x~, x~ + x~, x j ,

which is polynomial as well as its inverse. We can check that for every 2 c, 121r 1, the function

x~ x~ 222,x~ h;,(x)= xl 2 - 1 ' x 2 2 - 1 ~ - ( 2 - 1 ) ( 2 2 - 1 )

;~( 1 + 22)x 4 "~

- (2 - 1) (2 2 - 1 ) (2 3 - 1 ) ' X3 J is such that s 2s for all x e C 3. Also s is bijective and polynomial as well as its inverse.

A c k n o w l e d g e m e n t s

Gaetano Zampieri was supported by the Consiglio Nazionale delle Ricerche (215.26/01 CNR-NATO).

882 B. D e n g et al. Z A M P

We thank Arno van den Essen who showed us that h can fail to be a polynomial function in the case of Keller's functions of the homogeneous form.

References

[ 1] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Vet- lag, Berlin 1983.

[2] H. Bass, E. Connell, and D. Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7, 287 330 (1982).

[3] A. Biatynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13, 200-203 (1962).

[4] G. De Marco, G. Gorni, and G. Zampieri, Global inversion of functions: an introduction, Nonlinear Diff. Equat. and Appl. I, 229-248 (1994).

[5] L. Dru2kowski, An effective approach to Keller's Jacobian conjecture, Math. Ann. 264, 303-313 (1983).

[6] S. Friedland and J. Milnor, Plane polynomial automorphisms, Ergod. Th. and Dynam. Syst. 9, 67-99 (1989).

[7] O. H. Keller, Ganze Cremona trasformationen, Monatshefte ffir Mathematik und Physik 47, 299-306 (1939).

[8] G. H. Meisters, Inverting polynomial maps of n-space by solving differential equations, in Fink, Miller, Kliemann, eds., Delay and Differential Equations: Proc. in Honor of George Seifert on his retirement, World Sci. Publ. Co., 107-166 (1992).

[9] G. H. Meisters and C. Olech, Global stability, injectivity, and the Jacobian conjecture, de Gruyter, Berlin 1994. Proc. First World Congress of Nonlinear Analysts, Tampa, Florida. (Ed. Lakshmikan- tham).

[10] D. J. Newman, One-one polynomial maps, Proc. Amer. Math. Soc. 11, 867 870 (1960). [11] K. Rusek, A geometric approach to Keller's Jacobian conjecture, Math. Ann. 264, 315-320 (1983). [12] J. Sotomayor, Inversion of smooth mappings, Z. angew. Math. Phys. ZAMP 41, 306 310 (1990). [13] A. V. Yagzhev, Keller's problem, Siberian Math. J. 21, 747 754 (1980).

Abstract

We consider Keller's functions, namely polynomial functions f: C n -~ C n with det f'(x) = 1 at all x s C". Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open.

Without loss of generality assume f(0) = 0 andfr(0) = L We study the existence of certain mappings h~, 2 > 1, defined by power series in a ball with center at the origin, such that h~(0)= I and h~(2f(x)) = ;~h~(x). So each h~ conjugates 2f to its linear part 21 in a ball where it is injective.

We conjecture that for Keller's functions f of the homogeneous form

f(x) = x + g(x), g(sx) = sag(x), g '(x) ~ = O, x ~ C n, s ~ C

the conjugation h~ for 2f is an entire function.

(Received: October 22, 1994; revised: June 7, 1995)

Conjugation for polynomial mappings

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