Abstract. Let F = X + H be a cubic homogeneous polynomial automorphism from C

ANNALES

POLONICI MATHEMATICI LXII.2 (1995)

A counterexample to

a conjecture of Dru ˙zkowski and Rusek by Arno van den Essen (Nijmegen)

Abstract. Let F = X + H be a cubic homogeneous polynomial automorphism from C

ⁿ

to C

ⁿ

. Let p be the nilpotence index of the Jacobian matrix J H. It was conjectured by Dru˙zkowski and Rusek in [4] that deg F

⁻¹

≤ 3

^p−1

. We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.

1. Introduction. In [1] and [7] it was shown that it suffices to prove the Jacobian Conjecture for cubic homogeneous polynomial maps from C

ⁿ

to C

ⁿ

, i.e. maps of the form F = (F

1

, . . . , F

n

) with F

i

= X

i

+ H

i

, where each H

i

is either zero or a homogeneous polynomial of degree 3. In [2] it was shown that it even suffices to consider cubic linear polynomial maps, i.e. maps such that each H

i

is of the form H

i

= l

³_i

, where l

i

is a linear form.

A crucial result (cf. [1] and [6]) asserts that the degree of the inverse of a polynomial automorphism F is bounded by (deg F )

ⁿ⁻¹

(where deg F = max deg F

i

). In [4] Dru˙zkowski and Rusek proved that for cubic homoge- neous (resp. cubic linear) automorphisms this degree estimate could be im- proved in some special cases; more precisely, if ind J H denotes the index of nilpotency of J H then they showed that deg F

⁻¹

≤ 3

^{ind J H−1}

if ind J H ≤ 2 and also if H is cubic linear and ind J H ≤ 3. This led them to the following conjecture:

Conjecture 1.1 (D–R) ([4], 1985). If F = X+H is a cubic homogeneous polynomial automorphism, then deg F

⁻¹

≤ 3

^p−1

, where p = ind J H.

Recently, in [3], Dru˙zkowski showed that Conjecture D–R is true in case all coefficients of H are real numbers ≤ 0 (in which case the map F is stably tame, a result obtained by Yu in [8]).

In the present paper we show that the conjecture is true if n ≤ 4 and false if n ≥ 5.

1991 Mathematics Subject Classification: Primary 14E02.

Key words and phrases: polynomial automorphisms, Jacobian Conjecture.

[173]

174 A. v a n d e n E s s e n

2. The counterexample for n ≥ 5. Let n ≥ 5 and consider the poly- nomial ring C[X] := C[X

1

, . . . , X

n

].

Theorem 2.1. For each n ≥ 5 the polynomial automorphism

F = (X

1

+ 3X

₄²

X

2

− 2X

₄

X

5

X

3

, X

2

+ X

₄²

X

5

, X

3

+ X

₄³

, X

4

+ X

₅³

, X

5

, . . . , X

n

) is a counterexample to Conjecture D–R.

P r o o f. Put H = F − X. Then one easily verifies that (J H)

³

= 0 and (J H)

²

6= 0. Thus ind J H = 3. So if Conjecture D–R is true, then deg F

⁻¹

≤ 9. However, the inverse G = (G

₁

, . . . , G

n

) of F is given by the following formulas:

G

1

= X

1

− 3(X

₄

− X

₅³

)

²

(X

2

− (X

₄

− X

₅³

)

²

X

5

) + 2(X

4

− X

₅³

)X

5

(X

3

− (X

₄

− X

₅³

)

³

), G

2

= X

2

− (X

₄

− X

₅³

)

²

X

5

,

G

3

= X

3

− (X

₄

− X

₅³

)

³

, G

4

= X

4

− X

₅³

,

G

i

= X

i

for all 5 ≤ i ≤ n.

So looking at the highest power of X

5

appearing in G

1

, one easily verifies that deg G

1

= 13 > 9.

3. The case n ≤ 4. The main result of this section is Proposition 3.1. Conjecture D–R is true if n ≤ 4.

To prove this result we need the following theorem (cf. [5]):

Theorem 3.2. Let K be a field of characteristic zero and F = X − H a cubic homogeneous polynomial map in dimension four such that Det(J F )

= 1. Then there exists some T ∈ GL

4

(K) such that T

⁻¹

F T is of one of the following forms:









 x

1

x

2

x

3

x

4

− a

₄

x

³₁

− b

₄

x

²₁

x

2

− c

₄

x

²₁

x

3

− e

₄

x

1

x

²₂

− f

₄

x

1

x

2

x

3

− h

4

x

1

x

²₃

− k

4

x

³₂

− l

4

x

²₂

x

3

− n

4

x

2

x

²₃

− q

4

x

³₃









 , (1)





 x

1

x

2

−

¹₃

x

³₁

− h

₂

x

1

x

²₃

− q

₂

x

³₃

x

3

x

4

− x

²₁

x

3

− h

4

x

1

x

²₃

− q

4

x

³₃





 ,

(2)

A conjecture of Dru˙zkowski and Rusek 175

















 x

1

x

2

−

¹₃

x

³₁

− c

₁

x

²₁

x

4

+ 3c

1

x

1

x

2

x

3

−

^16q_48c^4c212^−r24 1

x

1

x

²₃

−

¹₂

r

4

x

1

x

3

x

4

+

³₄

r

4

x

2

x

²₃

−

^r_12c^4q4

1

x

³₃

−

_16c^r24

1

x

²₃

x

4

x

3

x

4

− x

²₁

x

3

+

_4c^r4

1

x

1

x

²₃

− 3c

₁

x

1

x

3

x

4

+ 9c

1

x

2

x

²₃

− q

₄

x

³₃

−

³₄

r

4

x

²₃

x

4

















 , (3)





 x

1

x

2

−

¹₃

x

³₁

x

3

− x

²₁

x

2

− e

3

x

1

x

²₂

− k

3

x

³₂

x

4

− e

₄

x

1

x

²₂

− k

₄

x

³₂





 , (4)













 x

1

x

2

−

¹₃

x

³₁

+ i

3

x

1

x

2

x

4

− j

₂

x

1

x

²₄

s

3

x

2

x

²₄

+ i

²₃

x

3

x

²₄

− t

₂

x

³₄

x

3

− x

²₁

x

2

−

^2s_i³

3

x

1

x

2

x

4

− i

3

x

1

x

3

x

4

− j

3

x

1

x

²₄

−

^s_i2²³ 3

x

2

x

²₄

− s

3

x

3

x

²₄

− t

3

x

³₄

x

4













 , (5)









 x

1

x

2

−

¹₃

x

³₁

− j

₂

x

1

x

²₄

− t

₂

x

³₄

x

3

− x

²₁

x

2

− e

3

x

1

x

²₂

− g

3

x

1

x

2

x

4

− j

3

x

1

x

²₄

− k

3

x

³₂

− m

₃

x

²₂

x

4

− p

₃

x

2

x

²₄

− t

₃

x

³₄

x

4









 , (6)









 x

1

x

2

−

¹₃

x

³₁

x

3

− x

²₁

x

2

− e

₃

x

1

x

²₂

− k

₃

x

³₂

x

4

− x

²₁

x

3

− e

₄

x

1

x

²₂

− f

₄

x

1

x

2

x

3

− h

₄

x

1

x

²₃

− k

₄

x

³₂

− l

4

x

²₂

x

3

− n

4

x

2

x

²₃

− q

4

x

³₃









 , (7)













 x

1

x

2

−

¹₃

x

³₁

x

3

− x

²₁

x

2

− e

₃

x

1

x

²₂

+ g

4

x

1

x

2

x

3

− k

₃

x

³₂

+ m

4

x

²₂

x

3

+ g

²₄

x

²₂

x

4

x

4

− x

²₁

x

3

− e

₄

x

1

x

²₂

−

^2m_g ⁴

4

x

1

x

2

x

3

− g

₄

x

1

x

2

x

4

− k

₄

x

³₂

−

^m_g2²⁴ 4

x

²₂

x

3

− m

₄

x

²₂

x

4













 . (8)

P r o o f. See [5, Theorem 2.7].

P r o o f o f 3.1. As remarked in the introduction, the case ind J H = n

was proved in [1] and [6]. The case ind J H = 2 was done in [4]. So we

may assume that 2 < ind J H < n. Therefore only the case n = 4 and

ind J H = 3 remains. By the classification theorem of Hubbers ([5, Theo-

rem 2.7]) we know that there exists T ∈ GL

4

(C) such that T

⁻¹

F T has one of

the eight forms described above. One easily verifies that in each of the eight

176 A. v a n d e n E s s e n

cases in which the nilpotency index of J H equals 3, deg(T

⁻¹

F T )

⁻¹

≤ 9, so deg F

⁻¹

≤ 9.

References

[1] H. B a s s, E. C o n n e l l, and D. W r i g h t, The Jacobian Conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982), 287–330.

[2] L. M. D r u ˙z k o w s k i, An effective approach to Keller’s Jacobian Conjecture, Math.

Ann. 264 (1983), 303–313.

[3] —, The Jacobian Conjecture: some steps towards solution, in: Automorphisms of Affine Spaces, Proc. Conf. “Invertible Polynomial Maps”, Cura¸ cao, July 4–8, 1994, A. R. P. van den Essen (ed.), Caribbean Mathematics Foundation, Kluwer Academic Publishers, 1995, 41–54.

[4] L. M. D r u ˙z k o w s k i and K. R u s e k, The formal inverse and the Jacobian conjecture, Ann. Polon. Math. 46 (1985), 85–90.

[5] E.-M. G. M. H u b b e r s, The Jacobian Conjecture: cubic homogeneous maps in dimen- sion four , master thesis, Univ. of Nijmegen, February 17, 1994; directed by A. R. P.

van den Essen.

[6] K. R u s e k and T. W i n i a r s k i, Polynomial automorphisms of C

ⁿ

, Univ. Iagel. Acta Math. 24 (1984), 143–149.

[7] A. V. Y a g z h e v, On Keller’s problem, Siberian Math. J. 21 (1980), 747–754.

[8] J.-T. Y u, On the Jacobian Conjecture: reduction of coefficients, J. Algebra 171 (1995), 515–523.

DEPARTMENT OF MATHEMATICS UNIVERSITY OF NIJMEGEN NIJMEGEN, THE NETHERLANDS E-mail: ESSEN@SCI.KUN.NL

Re¸ cu par la R´ edaction le 1.2.1995