Journal of Pure and Applied Algebra 216 (2012) 2709–2713
Contents lists available at SciVerse ScienceDirect
Journal of Pure and Applied Algebra
journal homepage: www.elsevier.com/locate/jpaa
Endomorphisms of polynomial algebras with small co-invariants
Xiaosong Sun
∗
School of Mathematics, Jilin University, Changchun, China
School of Mathematical Sciences, Peking University, Beijing, China
a r t i c l e i n f o
Article history:
Received 29 May 2011
Received in revised form 19 January 2012
Available online 7 May 2012
Communicated by J. Huebschmann
MSC: 14R10; 14R15
a b s t r a c t
Let φ be an endomorphism of the polynomial algebra C[x
1
, . . . , x
n
] and let co(φ) be
the dimension of the smallest sub-coalgebra C(φ) of the Hopf algebra C[G
n
a
] containing
φ(x
1
), . . . , φ(x
n
). We show that φ is a tame automorphism if det J(φ) ∈ C
∗
and one of the
following conditions is satisfied: (i) co(φ) ≤ n + 2; (ii) φ is quadratic and co(φ) ≤ n + 5.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
Let φ be an endomorphism of the polynomial algebra C[x
1
, . . . , x
n
]. We write φ = (φ(x
1
), . . . , φ(x
n
)), since φ is
determined by the φ(x
i
). The composition of two endomorphisms φ = (φ
1
, . . . , φ
n
) and ψ = (ψ
1
, . . . , ψ
n
) is ψ ◦ φ =
φ
1
(ψ
1
, . . . , ψ
n
), . . . , φ
n
(ψ
1
, . . . , ψ
n
)
. The degree of φ is defined by deg φ = max
1≤i≤n
deg φ(x
i
) and the Jacobian by
J(φ) =
∂φ(x
i
)/∂x
j
1≤i,j≤n
. The Jacobian conjecture asserts that φ is an automorphism if det J(φ) ∈ C
∗
; see [2] or
[4, introduction]. This conjecture is still open for any n ≥ 2.
We will refer to the number n as the dimension. An automorphism of the form (x
1
, . . . , x
i−1
, cx
i
+ f , x
i+1
, . . . , x
n
) is
called elementary if 0 ̸= c ∈ C and f ∈ C[x
1
, . . . ,
ˆ
x
i
, . . . , x
n
]. An automorphism is called tame if it is a finite composition
of elementary ones. The Tame Generators Problem asks whether every automorphism of C[x
1
, . . . , x
n
] is tame; see [4] or
[1]. In 1942, Jung [6] showed that the problem has an affirmative answer in dimension 2 for any field of characteristic zero
and, in 1953, van der Kulk [7], established the same kind of result in positive characteristic. In 1972, Nagata [12] offered a
candidate for a counterexample in dimension 3. Nagata’s conjecture was finally verified by Shestakov and Umirbaev [13] in
2004; in fact they showed that tame automorphisms of dimension 3 are algorithmically recognizable. However, the Tame
Generators Problem is still open in dimension n ≥ 4.
In 1994, Derksen showed that in dimension n ≥ 3, every tame automorphism is a finite composition of affine ones (i.e.
those of degree 1) and quadratic ones (i.e. those of degree 2); see [4, Theorem 5.2.1]. Rusek conjectured that every quadratic
automorphism is tame; see [4, Section 5.2]. The Rusek Conjecture was only verified for dimension n ≤ 5. (The case n ≤ 4 was
solved by Meisters and Olech [10] in 1991, and the case n = 5 was solved by de Bondt [3, Theorem 4.6.8] and independently
by the author [14, Theorem 4.2.3] in 2009.) The Rusek Conjecture is still open in dimension n ≥ 6.
It is well-known that C[x
1
, . . . , x
n
] = C[G
n
a
] has a Hopf algebra structure induced by the comultiplication ∆(x
i
) =
x
i
⊗ 1 + 1 ⊗ x
i
, with counit ϵ(x
i
) = 0 and antipode S(x
i
) = −x
i
, where G
a
denotes the additive group scheme; see for
example [17, Chapter 1].
In 1997, Le Bruyn [9] defined the co-invariant co(φ) of an endomorphism φ of C[x
1
, . . . , x
n
] to be the dimension
of the smallest sub-coalgebra C(φ) of the Hopf algebra C[G
n
a
] containing φ(x
i
), i = 1, . . . , n. One may observe that
co(φ) ≤
n+d−1
n
+ m(φ), where d = deg φ and m(φ) is the number of φ(x
i
) with deg φ(x
i
) = d. And co(φ) ≥ n + 1
∗
Correspondence to: School of Mathematics, Jilin University, Changchun, China.
E-mail address: sunxs@jlu.edu.cn.
0022-4049/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.jpaa.2012.03.030