Two Results on Homogeneous Hessian Nilpotent Polynomials
TWO RESULTS ON HOMOGENEOUS HESSIAN
NILPOTENT POLYNOMIALS
ARNO VAN DEN ESSEN∗ AND WENHUA ZHAO∗∗
Abstract. Let z = (z1, · · · , zn) and ∆ =
the Laplace
operator. A formal power series P (z) is said to be Hessian Nilpo-
tent(HN) if its Hessian matrix HesP (z) = ( ∂
∂zi∂zj
) is nilpotent. In
recent developments in [BE1], [M] and [Z], the Jacobian conjec-
ture has been reduced to the following so-called vanishing conjec-
ture(VC) of HN polynomials: for any homogeneous HN polynomial
P (z) (of degree d = 4), we have ∆mPm+1(z) = 0 for any m >> 0.
In this paper, we first show that, the VC holds for any homogeneous
HN polynomial P (z) provided that the projective subvarieties ZP
and Zσ2 of CP
n−1 determined by the principal ideals generated by
P (z) and σ2(z) :=
, respectively, intersect only at regular
points of ZP . Consequently, the Jacobian conjecture holds for the
symmetric polynomial maps F = z−∇P with P (z) HN if F has no
non-zero fixed point w ∈ Cn with
= 0. Secondly, we show
that the VC holds for a HN formal power series P (z) if and only
if, for any polynomial f(z), ∆m(f(z)P (z)m) = 0 when m >> 0.
1. Introduction and Main Results
Let z = (z1, z2, · · · , zn) be commutative free variables. Recall that
the well-known Jacobian conjecture claims that: any polynomial map
F (z) : Cn → Cn with the Jacobian j(F )(z) ≡ 1 is an autompophism of
n and its inverse map must also be a polynomial map. Despite intense
study from mathematicians in more than sixty years, the conjecture is
still open even for the case n = 2. In 1998, S. Smale [S] included the
Jacobian conjecture in his list of 18 important mathematical problems
for 21st century. For more history and known results on the Jacobian
conjecture, see [BCW], [E] and references there.
Recently, M. de Bondt and the first author [BE1] and G. Meng [M]
independently made the following remarkable breakthrough on the Ja-
cobian conjecture. Namely, they reduced the Jacobian conjecture to
2000 Mathematics Subject Classification. 14R15, 31B05.
Key words and phrases. Hessian nilpotent polynomials, the vanishing conjecture,
symmetric polynomial maps, the Jacobian conjecture.
http://arxiv.org/abs/0704.1690v1
2 ARNO VAN DEN ESSEN AND WENHUA ZHAO
the so-called symmetric polynomial maps, i.e the polynomial maps of
the form F = z − ∇P , where ∇P := ( ∂P
, · · · , ∂P
), i.e. ∇P (z) is
the gradient of P (z) ∈ C[z].
For more recent developments on the Jacobian conjecture for sym-
metric polynomial maps, see [BE1]–[BE4].
Based on the symmetric reduction above and also the classical homo-
geneous reduction in [BCW] and [Y], the second author in [Z] further
reduced the Jacobian conjecture to the following so-called vanishing
conjecture.
Let ∆:=
the Laplace operator and call a formal power series
P (z) Hessian nilpotent(HN) if its Hessian matrix HesP (z) := ( ∂
∂zi∂zj
is nilpotent. It has been shown in [Z] that the Jacobian conjecture is
equivalent to
Conjecture 1.1. (Vanishing Conjecture of HN Polynomials)
For any homogeneous HN polynomial P (z) (of degree d = 4), we have
∆mPm+1 = 0 when m >> 0.
Note that, it has also been shown in [Z] that P (z) is HN if and only
if ∆mPm = 0 for m ≥ 1.
In this paper, we will prove the following two results on HN polyno-
mials.
Let P (z) be a homogeneous HN polynomial of degree d ≥ 3 and
σ2(z):=
i=1 z
i . We denote by ZP and Zσ2 the projective subvarieties
of CP n−1 determined by the principal ideals generated by P (z) and
σ2(z), respectively. The first main result of this paper is the following
theorem.
Theorem 1.2. Let P (z) be a homogeneous HN polynomial of degree
d ≥ 4. Assume that ZP intersects with Zσ2 only at regular points of
ZP , then the vanishing conjecture holds for P (z). In particular, the
vanishing conjecture holds if the projective variety ZP is regular.
Remark 1.3. Note that, when degP (z) = d = 2 or 3, the Jacobian
conjecture holds for the symmetric polynomial map F = z −∇P . This
is because, when d = 2, F is a linear map with j(F ) ≡ 1. Hence F
is an automorphism of Cn; while when d = 3, we have degF = 2. By
Wang’s theorem [W], the Jacobian conjecture holds for F again. Then,
by the equivalence of the vanishing conjecture for the homogeneous HN
polynomial P (z) and the Jacobian conjecture for the symmetric map
F = z −∇P established in [Z], we see that, when deg P (z) = d = 2 or
3, Theorem 1.2 actually also holds even without the condition on the
projective variety ZP .
HOMOGENEOUS HESSIAN NILPOTENT POLYNOMIALS 3
For any non-zero z ∈ Cn, denote by [z] its image in the projective
space CP n−1. Set
Z̃σ2 := {z ∈ C
n | z 6= 0; [z] ∈ Zσ2}.(1.1)
In other words, Z̃σ2 is the set of non-zero z ∈ C
n such that
i=1 z
i = 0.
Note that, for any homogeneous polynomial P (z) of degree d, it
follows from the Euler’s formula dP =
i=1 zi
, that any non-zero
w ∈ Cn, [w] ∈ CP n−1 is a singular point of ZP if and only if w is a
fixed point of the symmetric map F = z−∇P . Furthermore, it is also
well-known that, j(F ) ≡ 1 if and only if P (z) is HN.
By the observations above and Theorem 1.2, it is easy to see that
we have the following corollary on symmetric polynomial maps.
Corollary 1.4. Let F = z − ∇P with P homogeneous and j(F ) ≡ 1
(or equivalently, P is HN). Assume that F does not fix any w ∈ Z̃σ2 .
Then the Jacobian holds for F (z). In particular, if F has no non-zero
fixed point, the Jacobian conjecture holds for F .
Our second main result is following theorem which says that the
vanishing conjecture is actually equivalent to a formally much stronger
statement.
Theorem 1.5. For any HN polynomial P (z), the vanishing conjec-
ture holds for P (z) if and only if, for any polynomial f(z) ∈ C[z],
∆m(f(z)P (z)m) = 0 when m >> 0.
2. Proof of the Main Results
Let us first fix the following notation. Let z = (z1, z2, · · · , zn) be
free complex variables and C[z] (resp.C[[z]]) the algebra of polynomials
(resp. formal power series) in z. For any d ≥ 0, we denote by Vd the
vector space of homogeneous polynomials in z of degree d.
For any 1 ≤ i ≤ n, we set Di =
and D = (D1, D2, · · · , Dn). We
define a C-bilinear map {·, ·} : C[z]× C[z] → C[z] by setting
{f, g} := f(D)g(z)
for any f(z), g(z) ∈ C[z].
Note that, for any m ≥ 0, the restriction of {·, ·} on Vm × Vm gives
a C-bilinear form of the vector subspace Vm, which we will denote by
Bm(·, ·). It is easy to check that, for any m ≥ 1, Bm(·, ·) is symmetric
and non-singular.
The following lemma will play a crucial role in our proof of the first
main result.
4 ARNO VAN DEN ESSEN AND WENHUA ZHAO
Lemma 2.1. For any homogeneous polynomials gi(z) (1 ≤ i ≤ k) of
degree di ≥ 1, let S be the vector space of polynomial solutions of the
following system of PDEs:
g1(D) u(z) = 0,
g2(D) u(z) = 0,
.....
gk(D) u(z) = 0.
(2.2)
Then, dimS < +∞ if and only if gi(z) (1 ≤ i ≤ k) have no non-zero
common zeroes.
Proof: Let I the homogeneous ideal of C[z] generated by {gi(z)|1 ≤
i ≤ k}. Since all gi(z)’s are homogeneous, S is a homogeneous vector
subspace S of C[z].
Write
Sm,(2.3)
Im.(2.4)
where Im := I ∩ Vm and Sm := I ∩ Vm for any m ≥ 0.
Claim: For any m ≥ 1 and u(z) ∈ Vm, u(z) ∈ Sm if and only if
{u, Im} = 0, or in other words, Sm = I
m with respect to the C-bilinear
form Bm(·, ·) of Vm.
Proof of the Claim: First, by the definitions of I and S, we have
{Im, Sm} = 0 for any m ≥ 1, hence Sm ⊆ I
m. Therefore, we need only
show that, for any u(z) ∈ I⊥m ⊂ Vm, gi(D)u(z) = 0 for any 1 ≤ i ≤ n.
We first fix any 1 ≤ i ≤ n. If m < di, there is nothing to prove. If
m = di, then gi(z) ∈ Im, hence {gi, u} = gi(D)u = 0. Now suppose
m > di. Note that, for any v(z) ∈ Vm−di , v(z)gi(z) ∈ Im. Hence we
0 = {v(z)gi(z), u(z)}
= v(D)gi(D)u(z)
= v(D) (gi(D)u) (z)
= {v(z), (gi(D)u) (z)}.
Therefore, we have
Bm−di ((gi(D)u)(z), Vm−di) = 0.
HOMOGENEOUS HESSIAN NILPOTENT POLYNOMIALS 5
Since Bm−di(·, ·) is a non-singular C-bilinear form of Vm−di , we have
gi(D)u = 0. Hence, the Claim holds. ✷
By a well-known fact in Algebraic Geometry (see Exercise 2.2 in
[H], for example), we know that the homogeneous polynomials gi(z)
(1 ≤ i ≤ k) have no non-zero common zeroes if and only if Im = Vm
when m >> 0. While, by the Claim above, we know that, Im = Vm
when m >> 0 if and only if Sm = 0 when m >> 0, and if and only if
the solution space S of the system (2.2) is finite dimensional. Hence,
the lemma follows. ✷
Now we are ready to prove our first main result, Theorem 1.2.
Proof of Theorem 1.2: Let P (z) be a homogeneous HN polynomial
of degree d ≥ 4 and S the vector space of polynomial solutions of the
following system of PDEs:


(D) u(z) = 0,
(D) u(z) = 0,
.....
(D) u(z) = 0,
∆ u(z) = 0.
(2.5)
First, note that the projective subvariety ZP intersects with Zσ2 only
at regular points of ZP if and only if
(z) (1 ≤ i ≤ n) and σ2 =∑n
i=1 z
i have no non-zero common zeros (agian use Euler’s formula).
Then, by Lemma 2.1, we have dimS < +∞.
On the other hand, by Theorem 6.3 in [Z], we know that ∆mPm+1 ∈
S for any m ≥ 0. Note that deg∆mPm+1 = (d−2)m+d for any m ≥ 0.
So deg∆mPm+1 > deg∆kP k+1 for any m > k. Since dimS < +∞
(from above), we have ∆mPm+1 = 0 when m >> 0, i.e. the vanishing
conjecture holds for P (z). ✷
Next, we give a proof for our second main result, Theorem 1.5.
Proof of Theorem 1.5: The (⇐) part follows directly by choosing
f(z) to be P (z) itself.
To show (⇒) part, let d = deg f(z). If d = 0, f is a constant. Then,
∆m(f(z)P (z)m) = f(z)∆mPm = 0 for any m ≥ 1.
So we assume d ≥ 1. By Theorem 6.2 in [Z], we know that, if
the vanishing conjecture holds for P (z), then, for any fixed a ≥ 1,
∆mPm+a = 0 when m >> 0. Therefore there exists N > 0 such that,
for any 0 ≤ b ≤ d and any m > N , we have ∆mPm+b = 0.
6 ARNO VAN DEN ESSEN AND WENHUA ZHAO
By Lemma 6.5 in [Z], for any m ≥ 1, we have
∆m(f(z)P (z)m) =(2.6)
k1+k2+k3=m
k1,k2,k3≥0
k1, k2, k3
|s|=k2
∂k2∆k1f(z)
∂k2∆k3Pm(z)
where
k1,k2,k3
denote the usual binomials.
Note first that, the general term in the sum above is non-zero only
if 2k1 + k2 ≤ d. But on the other hand, since
0 ≤ k1 + k2 ≤ 2k1 + k2 ≤ d,(2.7)
by the choice of N ≥ 1, we have ∆k3Pm(z) = ∆k3P k3+(k1+k2)(z) is
non-zero only if
k3 ≤ N.(2.8)
From the observations above and Eqs. (2.6), (2.7), (2.8) it is easy to
see that, ∆m(f(z)P (z)m) 6= 0 only if m = k1 + k2 + k3 ≤ d + N . In
other words, ∆m(f(z)P (z)m) = 0 for any m > d+N . Hence Theorem
1.5 holds. ✷
Note that all results used in the proof above for the (⇐) part of the
theorem also hold for all HN formal power series. Therefore we have
the following corollary.
Corollary 2.2. Let P (z) be a HN formal power series such that the
vanishing conjecture holds for P (z). Then, for any polynomial f(z),
we have ∆m(f(z)P (z)m) = 0 when m >> 0.
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∗ Department of Mathematics, Radboud University Nijmegen,
Postbus 9010, 6500 GL Nijmegen, The Netherlands.
E-mail: essen@math.ru.nl
∗∗ Department of Mathematics, Illinois State University, Nor-
mal, IL 61790-4520.
E-mail: wzhao@ilstu.edu.
http://arxiv.org/abs/math-ph/0308035
http://arxiv.org/abs/math/0409534
	1. Introduction and Main Results
	2. Proof of the Main Results
	References
