### 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

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

^{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

_{4}

^{2}

## X

2## − 2X

_{4}

## X

5## X

3## , X

2## + X

_{4}

^{2}

## X

5## , X

3## + X

_{4}

^{3}

## , X

4## + X

_{5}

^{3}

## , 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

_{5}

^{3}

## )

^{2}

## (X

2## − (X

_{4}

## − X

_{5}

^{3}

## )

^{2}

## X

5## ) + 2(X

4## − X

_{5}

^{3}

## )X

5## (X

3## − (X

_{4}

## − X

_{5}

^{3}

## )

^{3}

## ), G

2## = X

2## − (X

_{4}

## − X

_{5}

^{3}

## )

^{2}

## X

5## ,

## G

3## = X

3## − (X

_{4}

## − X

_{5}

^{3}

## )

^{3}

## , G

4## = X

4## − X

_{5}

^{3}

## ,

## 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

^{3}

_{1}

## − b

_{4}

## x

^{2}

_{1}

## x

2## − c

_{4}

## x

^{2}

_{1}

## x

3## − e

_{4}

## x

1## x

^{2}

_{2}

## − f

_{4}

## x

1## x

2## x

3## − h

4## x

1## x

^{2}

_{3}

## − k

4## x

^{3}

_{2}

## − l

4## x

^{2}

_{2}

## x

3## − n

4## x

2## x

^{2}

_{3}

## − q

4## x

^{3}

_{3}

##

##

##

##

## , (1)

##

##

## x

1## x

2## −

^{1}

_{3}

## x

^{3}

_{1}

## − h

_{2}

## x

1## x

^{2}

_{3}

## − q

_{2}

## x

^{3}

_{3}

## x

3## x

4## − x

^{2}

_{1}

## x

3## − h

4## x

1## x

^{2}

_{3}

## − q

4## x

^{3}

_{3}

##

##

## ,

## (2)

*A conjecture of Dru˙zkowski and Rusek* 175

##

##

##

##

##

##

##

##

## x

1## x

2## −

^{1}

_{3}

## x

^{3}

_{1}

## − c

_{1}

## x

^{2}

_{1}

## x

4## + 3c

1## x

1## x

2## x

3## −

^{16q}

_{48c}

^{4}

^{c}

^{2}

^{1}2

^{−r}

^{2}

^{4}1

## x

1## x

^{2}

_{3}

## −

^{1}

_{2}

## r

4## x

1## x

3## x

4## +

^{3}

_{4}

## r

4## x

2## x

^{2}

_{3}

## −

^{r}

_{12c}

^{4}

^{q}

^{4}

1

## x

^{3}

_{3}

## −

_{16c}

^{r}

^{2}

^{4}

1

## x

^{2}

_{3}

## x

4## x

3## x

4## − x

^{2}

_{1}

## x

3## +

_{4c}

^{r}

^{4}

1

## x

1## x

^{2}

_{3}

## − 3c

_{1}

## x

1## x

3## x

4## + 9c

1## x

2## x

^{2}

_{3}

## − q

_{4}

## x

^{3}

_{3}

## −

^{3}

_{4}

## r

4## x

^{2}

_{3}

## x

4##

##

##

##

##

##

##

##

## , (3)

##

##

## x

1## x

2## −

^{1}

_{3}

## x

^{3}

_{1}

## x

3## − x

^{2}

_{1}

## x

2## − e

3## x

1## x

^{2}

_{2}

## − k

3## x

^{3}

_{2}

## x

4## − e

_{4}

## x

1## x

^{2}

_{2}

## − k

_{4}

## x

^{3}

_{2}

##

##

## , (4)

##

##

##

##

##

##

## x

1## x

2## −

^{1}

_{3}

## x

^{3}

_{1}

## + i

3## x

1## x

2## x

4## − j

_{2}

## x

1## x

^{2}

_{4}

## s

3## x

2## x

^{2}

_{4}

## + i

^{2}

_{3}

## x

3## x

^{2}

_{4}

## − t

_{2}

## x

^{3}

_{4}

## x

3## − x

^{2}

_{1}

## x

2## −

^{2s}

_{i}

^{3}

3

## x

1## x

2## x

4## − i

3## x

1## x

3## x

4## − j

3## x

1## x

^{2}

_{4}

## −

^{s}

_{i}2

^{2}

^{3}3

## x

2## x

^{2}

_{4}

## − s

3## x

3## x

^{2}

_{4}

## − t

3## x

^{3}

_{4}

## x

4##

##

##

##

##

##

## , (5)

##

##

##

##

## x

1## x

2## −

^{1}

_{3}

## x

^{3}

_{1}

## − j

_{2}

## x

1## x

^{2}

_{4}

## − t

_{2}

## x

^{3}

_{4}

## x

3## − x

^{2}

_{1}

## x

2## − e

3## x

1## x

^{2}

_{2}

## − g

3## x

1## x

2## x

4## − j

3## x

1## x

^{2}

_{4}

## − k

3## x

^{3}

_{2}

## − m

_{3}

## x

^{2}

_{2}

## x

4## − p

_{3}

## x

2## x

^{2}

_{4}

## − t

_{3}

## x

^{3}

_{4}

## x

4##

##

##

##

## , (6)

##

##

##

##

## x

1## x

2## −

^{1}

_{3}

## x

^{3}

_{1}

## x

3## − x

^{2}

_{1}

## x

2## − e

_{3}

## x

1## x

^{2}

_{2}

## − k

_{3}

## x

^{3}

_{2}

## x

4## − x

^{2}

_{1}

## x

3## − e

_{4}

## x

1## x

^{2}

_{2}

## − f

_{4}

## x

1## x

2## x

3## − h

_{4}

## x

1## x

^{2}

_{3}

## − k

_{4}

## x

^{3}

_{2}

## − l

4## x

^{2}

_{2}

## x

3## − n

4## x

2## x

^{2}

_{3}

## − q

4## x

^{3}

_{3}

##

##

##

##

## , (7)

##

##

##

##

##

##

## x

1## x

2## −

^{1}

_{3}

## x

^{3}

_{1}

## x

3## − x

^{2}

_{1}

## x

2## − e

_{3}

## x

1## x

^{2}

_{2}

## + g

4## x

1## x

2## x

3## − k

_{3}

## x

^{3}

_{2}

## + m

4## x

^{2}

_{2}

## x

3## + g

^{2}

_{4}

## x

^{2}

_{2}

## x

4## x

4## − x

^{2}

_{1}

## x

3## − e

_{4}

## x

1## x

^{2}

_{2}

## −

^{2m}

_{g}

^{4}

4

## x

1## x

2## x

3## − g

_{4}

## x

1## x

2## x

4## − k

_{4}

## x

^{3}

_{2}

## −

^{m}

_{g}2

^{2}

^{4}4

## x

^{2}

_{2}

## x

3## − m

_{4}

## x

^{2}

_{2}

## 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