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
nto 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
nto C
n, i.e. maps of the form F = (F
1, . . . , F
n) with F
i= X
i+ H
i, where each H
iis 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
iis of the form H
i= l
3i, where l
iis 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−1if 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
42X
2− 2X
4X
5X
3, X
2+ X
42X
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)
26= 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)
2X
5) + 2(X
4− X
53)X
5(X
3− (X
4− X
53)
3), G
2= X
2− (X
4− X
53)
2X
5,
G
3= X
3− (X
4− X
53)
3, G
4= X
4− X
53,
G
i= X
ifor all 5 ≤ i ≤ n.
So looking at the highest power of X
5appearing 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
−1F T is of one of the following forms:
x
1x
2x
3x
4− a
4x
31− b
4x
21x
2− c
4x
21x
3− e
4x
1x
22− f
4x
1x
2x
3− h
4x
1x
23− k
4x
32− l
4x
22x
3− n
4x
2x
23− q
4x
33
, (1)
x
1x
2−
13x
31− h
2x
1x
23− q
2x
33x
3x
4− x
21x
3− h
4x
1x
23− q
4x
33
,
(2)
A conjecture of Dru˙zkowski and Rusek 175
x
1x
2−
13x
31− c
1x
21x
4+ 3c
1x
1x
2x
3−
16q48c4c212−r24 1x
1x
23−
12r
4x
1x
3x
4+
34r
4x
2x
23−
r12c4q41
x
33−
16cr241
x
23x
4x
3x
4− x
21x
3+
4cr41
x
1x
23− 3c
1x
1x
3x
4+ 9c
1x
2x
23− q
4x
33−
34r
4x
23x
4
, (3)
x
1x
2−
13x
31x
3− x
21x
2− e
3x
1x
22− k
3x
32x
4− e
4x
1x
22− k
4x
32
, (4)
x
1x
2−
13x
31+ i
3x
1x
2x
4− j
2x
1x
24s
3x
2x
24+ i
23x
3x
24− t
2x
34x
3− x
21x
2−
2si33
x
1x
2x
4− i
3x
1x
3x
4− j
3x
1x
24−
si223 3x
2x
24− s
3x
3x
24− t
3x
34x
4
, (5)
x
1x
2−
13x
31− j
2x
1x
24− t
2x
34x
3− x
21x
2− e
3x
1x
22− g
3x
1x
2x
4− j
3x
1x
24− k
3x
32− m
3x
22x
4− p
3x
2x
24− t
3x
34x
4
, (6)
x
1x
2−
13x
31x
3− x
21x
2− e
3x
1x
22− k
3x
32x
4− x
21x
3− e
4x
1x
22− f
4x
1x
2x
3− h
4x
1x
23− k
4x
32− l
4x
22x
3− n
4x
2x
23− q
4x
33
, (7)
x
1x
2−
13x
31x
3− x
21x
2− e
3x
1x
22+ g
4x
1x
2x
3− k
3x
32+ m
4x
22x
3+ g
24x
22x
4x
4− x
21x
3− e
4x
1x
22−
2mg 44
x
1x
2x
3− g
4x
1x
2x
4− k
4x
32−
mg224 4x
22x
3− m
4x
22x
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
−1F 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
−1F T )
−1≤ 9, so deg F
−1≤ 9.
References
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[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.
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DEPARTMENT OF MATHEMATICS UNIVERSITY OF NIJMEGEN NIJMEGEN, THE NETHERLANDS E-mail: ESSEN@SCI.KUN.NL