Jacobian Conjecture and Nilpotent Mappings

Vik.S. Kulikov ^{∗}

Abstract

We prove the equivalence of the Jacobian Conjecture (JC(n)) and the Conjecture on the cardinality of the set of fixed points of a polynomial nilpotent mapping (JN(n)) and prove a series of assertions confirming JN(n).

Introduction

Let F : ℂ^{n} → ℂ^{n} be a morphism of complex affine spaces. If we choose coordinates in the image and preimage, then F is defined by n polynomials Y_{i} = F_{i}(X_{1}, ..., X_{n}) ∈ ℂ[X] = ℂ[X_{1}, ..., X_{n}] in n variables. Put

F′(X) = det JF (X),

where

JF = (\frac{∂F_{i}}{∂X_{j}} )_{1≤i,j≤n}

is the Jacobi martix of F.

It is well-known that F is locally invertible in a neighborhood of some point x ∈ ℂ^{n} (as an analytic mapping) if and only if F′(x) ≠ 0. Therefore, ”F′(x) ≠ 0 for all x ∈ ℂ^{n}” is a necessary condition for the invertibility of F.

The famous Jacobian Conjecture (JC(n)) claims: Conjecture JC(n). Let F : ℂ^{n} → ℂ^{n} be a morphism such that F′(x) ≠ 0 for all x ∈ ℂ^{n} (or, equivalently, F′(X) ∈ ℂ^{∗}), then F is an isomorphism, i.e. F possesses an inverse mapping which is also given by polynomials.

Without loss of generality we can assume that F′(X) ≡ 1 if F′(X) ∈ ℂ^{∗}. The following reformulations of conjecture JC(n) are well-known

Theorem 1 (cf. [2]). Let F : ℂ^{n} → ℂ^{n} be a morphism with F′(X) ∈ ℂ^{∗}. The Jacobian conjecture JC(n) is equivalent to either of the following assertions:

(i) F is injective;

(ii) The degree deg F of F is equal to 1 (i.e. ℂ(X_{1}, ..., X_{n})) = ℂ(F_{1}(X), ..., F_{n}(X))); (iii) The ring ℂ[X_{1}, ..., X_{n}] is a finitely generated ℂ[F_{1}(X), ..., F_{n}(X)]-module; (iv) The ring ℂ[X_{1}, ..., X_{n}] is a projective ℂ[F_{1}(X), ..., F_{n}(X)]-module; (v) The ring ℂ[F_{1}(X), ..., F_{n}(X)] is integrally closed in ℂ[X_{1}, ..., X_{n}]; (vi) The field C(X_{1}, ..., X_{n}) is a Galois extension over ℂ(F_{1}(X), ..., F_{n}(X)); (vii) F is proper.

∗Partly supported by RFFI (No. 96-01-00614) and INTAS (No. 96-0713).

The Jacobian Conjecture have been formulated by O.H.Keller in 1939 [7] in the case n = 2 for the polynomials with integer coefficients. For n = 1 conjecture JC(n) is trivial since a polynomial F (X_{1}) ∈ ℂ[X_{1}] with F′(X_{1}) ∈ ℂ^{∗} is of degree one and, therefore, defines an isomorphism F : ℂ^{1} → ℂ^{1}. The Jacobian Conjecture JC(n) remains an open problem even in the case n = 2 in spite of attempts of many mathematicians to prove it or to find a counterexample (a survey of the results on the Jacobian Conjecture one can find in [2], [4], [5]). These attempts allow to obtain many partial results connected with the Jacobian Conjecture and even several false proofs were published (for the critique of these false proofs we refer to [12] and [2] ).

The aim of this short note is to attract attention of the reader to a connection of JC(n) and one of conjectures on the nilpotent mappings.

Definition 1 A morphism N = (N_{1}, ..., N_{n}) : ℂ^{n} → ℂ^{n} is called nilpotent if JN(X) is a nilpotent matrix, i.e.

(JN(X))^{n} = (\frac{∂N_{i}}{∂X_{i}} )^{n} ≡ 0

in the ring of matrices M(n, ℂ[X]) of n-th order with coefficients in ℂ[X].

We consider nilpotent mappings as endomorphisms of ℂ^{n}. A point x ∈ ℂ^{n} is called a fixed point of a nilpotent mapping N if N(x) = x. It is well-known that if N is linear and nilpotent, then N^{n}(ℂ^{n}) consists of a unique point. In particular, a linear nilpotent mapping N possesses only one fixed point. Conjecture JN(n). A nilpotent mapping N : ℂ^{n} → ℂ^{n} possesses at most one fixed point.

The following theorem describes a connection between the conjectures JC(n) and JN(n).

Theorem 2 The conjectures JC(n) and JN(n) are equivalent, i.e. JC(n) is true for all n if and only if JN(n) is true for all n.

In section 1, we recall some well-known results relating to the Jacobian Conjecture. In section 2, we prove Theorem 2 and a series of assertions confirming JN(n).

This paper was written during my stay at the Max-Planck-Institut für Mathematik in Bonn. It is a pleasure to thank the Institut for its hospitality and financial support.

1 Stable equivalence of mappings and the Jacobian Conjecture

1.1. Let F : ℂ^{n} → ℂ^{n} and G : ℂ^{n} → ℂ^{n} be morphisms given by polynomials F_{1}(X), ..., F_{n}(X) and G_{1}(X), ..., G_{n}(X) respectively. We shall say that F and G are equivalent if there exist automorphisms H, R : ℂ^{n} → ℂ^{n} such that the following diagram is commutative

[figure: ℂ^{n}; G; ℂ^{n}; R; ℂ^{n}.; ℂ^{n}; H; F]

In other words, F and G are equivalent if

R_{i}(G_{1}(X), ..., G_{n}(X)) = F_{i}(H_{1}(X), ..., H_{n}(X))

for all i, where H_{1}(X), ..., H_{n}(X) and R_{1}(X), ..., R_{n}(X) are polynomials in ℂ[X_{1}, ..., X_{n}] defining the automorphisms H and R.

Definition 2 Morphisms F : ℂ^{n} → ℂ^{n} and G : ℂ^{m} → ℂ^{m}, n ≤ m, are called stably equivalent if F × Id : ℂ^{m} = ℂ^{n} × ℂ^{m−n} → ℂ^{n} × ℂ^{m−n} = ℂ^{m} and G are equivalent.

The following assertion is a consequence of the definition.

Claim 1 If F and G are stably equivalent, then

deg F = deg G.

In particular, if F′(X) ∈ ℂ^{∗} for F and F and G are stably equivalent, then F and G are invertible (or not invertible) simultaneously.

1.2. To avoid confusion with the degree of a morphism, we shall call p-degree or polynomial degree of a morphism F = (F_{1}, ..., F_{n}) : ℂ^{n} → ℂ^{n} the maximum of the degrees of F_{i}(X) defining F,

deg_{p} F = max deg F_{i}(X).

Of course, p-degree depends on a choice of affine coordinates in the image and preimage.

A usual construction to simplify the form of polynomials defining a morphism is the following: one replaces a mapping by a stably equivalent one and performs linear and triangular automorphisms of the image and the preimage.

Definition 3 An automorphism H : ℂ^{n} → ℂ^{n} is called triangular if, in some coordinate system, it can be given by polynomials H_{i}(X_{1}, ..., X_{n}), i = 1, ..., n, of the form

H_{i}(X_{1}, ..., X_{n}) = X_{i} + H_{i}(X_{1}, ..., X_{i−1}),

where H_{i}(X_{1}, ..., X_{i−1}) ∈ ℂ[X_{1}, ..., X_{i−1}] is a polynomial in variables X_{1}, ..., X_{i−1}.

The following theorem is well-known.

Theorem 3 ([2], [15]). For any morphism F : ℂ^{n} → ℂ^{n} there exists a stably equivalent morphism G : ℂ^{m} → ℂ^{m} such that deg_{p} G ≤ 3.

1.3. Let F : ℂ^{n} → ℂ^{n}, given by polynomials F_{i}(X), map the origin to the origin, i.e. F (0) = 0. A morphism F_{t} : ℂ^{n+1} → ℂ^{n+1} given by the polynomials

\begin{aligned}F̃_{i}(X_{1}, ..., X_{n}, T) & = & ∑_{j=1}^{deg F_{i}} T^{j−1}F_{(j)i}(X_{1}, ..., X_{n}), i = 1, ..., n, \\ F̃_{n+1}(X_{1}, ..., X_{n}, T) & = & T, \end{aligned}

is called a blow up of F, where F_{(j)i}(X) is the homogeneous component of degree j of the polynomial F_{i}(X). It is easy to check the following assertion.

Claim 2 Let F : ℂ^{n} → ℂ^{n} be a morphism such that F (0) = 0, then (i) F′_{t}(X_{1}, ..., X_{n}, T) = F′(T X_{1}, ..., T X_{n}); (ii) F, with F′(X) ∈ ℂ^{∗}, is invertible if and only if F_{t} is invertible.

Geometrically, to obtain F_{t} from F, one needs to perform σ-processes with centers at the origins of the image and the preimage of the morphism F × Id : ℂ^{n+1} = ℂ^{n} × ℂ^{1} → ℂ^{n} × ℂ^{1} = ℂ^{n+1} and to restrict the obtained morphism onto coordinates neighborhoods corresponding the last coordinate T.

In [3], theorem 3 was made more precise.

Theorem 4 Let F : ℂ^{n} → ℂ^{n} be a morphism such that F (0) = 0 and F′(0) ≠ 0. Then the blow up F_{t} of F is stably equivalent to G : ℂ^{m} → ℂ^{m} defined by polynomials G_{i}(X) of the form

G_{i}(X_{1}, ..., X_{m}) = X_{i} + G_{(3)i}(X_{1}, ..., X_{m}),  (1)

where G_{(3)i}(X_{1}, ..., X_{m}) are homogeneous forms of degree 3.

Since the space of the homogeneous polynomials of degree k is generated by k-th powers of the linear polynomials, the following theorem is a simple consequence of Theorem 4.

Theorem 5 Let F : ℂ^{n} → ℂ^{n} be given by polynomials

F_{i}(X_{1}, ..., X_{n}) = X_{i} + F_{(3)i}(X_{1}, ..., X_{n}).

Then F is stably equivalent to G : ℂ^{m} → ℂ^{m} given by polynomials G_{i}(X) of the form

G_{i}(X_{1}, ..., X_{m}) = X_{i} + (∑ a_{i,j}X_{j})^{3}.  (2)

1.4. The Jacobian conjecture was proved in [14] and [6] for the morphisms of the form (1) with m ≤ 4 and in [6] for the morphisms of the form (2) with m ≤ 7. Besides, in [13] the following theorem was proved

Theorem 6 The conjecture JC(n) is true for F : ℂ^{n} → ℂ^{n} of polynomial degree deg_{p} F ≤ 2.

This theorem will be obtained in the next section as a consequence of more general result (see Proposition 1).

Note also that the conjecture JC(n) was proved in [8] in the following two cases: when F is given by polynomials F_{i}(X) ∈ X_{i}ℂ[X] and when F_{i}(X) = X_{i} + λ_{i}M_{i}(X), where M_{i}(X) are monomials. 1.5. In the case F : ℂ^{2} → ℂ^{2} the Jacobian conjecture holds for all F of polynomial degree deg_{p} F ≤ 100 [10]. In [9] (see also [5]), it was proved that if the degrees of polynomials defining F with F′(X) ∈ ℂ^{∗} are coprime, then F is invertible and moreover in this case one of polynomials must be linear. In particular, the Jacobian conjecture holds if one of the polynomials defining F is of prime degree. In [1], this result was generalized to the case when the degree of one of the polynomials defining F is a product of two different prime integers.

In addition, in [11], it was shown that if F : ℂ^{2} → ℂ^{2} with F′(X) ≡ 1 is not invertible, then deg F ≥ 4. 1.6. A morphism F : ℂ^{n} → ℂ^{n} induces a polynomial mapping F̃ : ℝ^{2n} → ℝ^{2n} given by

F̃ = (Re F_{1}, Im F_{1}, ..., Re F_{n}, Im F_{n}),

where F_{i} are polynomials defining F. It is easy to see that

det JF̃ =| det JF |^{2} .

Hence, det JF ∈ ℂ^{∗} if and only if det JF̃ ∈ ℝ^{∗}. Conjecture RJC(n). A polynomial mapping F : ℝ^{n} → ℝ^{n} with det JF ∈ ℝ^{∗} is injective.

It is easy to show that if RJC(n) is true, then F : ℝ^{n} → ℝ^{n} with det JF ∈ ℝ^{∗} is invertible and the following proposition holds.

Proposition 1 The conjecture JC(n) is true for all n if and only if the conjecture RJC(n) is true for all n.

Consider F : ℝ^{n} → ℝ^{n} with det JF ∈ ℝ^{∗}. Without loss of generality, we can assume that F (0) = 0 and JF (0) = E, where E is the identity matrix. Write F in the form

F = Id + F_{(2)} + ... + F_{(m)},  (3)

where F_{(i)} are morphisms defined by homogeneous polynomials of degree i.

Definition 4 A polynomial mapping F : ℝ^{n} → ℝ^{n} of the form (3) is called positive (resp. negative) if all non-zero coefficients of the F_{(i)} are positive (resp. negative).

In [16], the following theorem has been proved.

Theorem 7 (i) The conjecture RJC(n) holds for all n and all negative polynomial mappings F : ℝ^{n} → ℝ^{n}.

(ii) It suffices to prove RJC(n) for all positive polynomial mappings F : ℝ^{n} → ℝ^{n}.

It follows from Theorem 7 (i) (passing to the blow up of F) that

The conjecture RJC(n) holds for all n and for all positive polynomials mappings F : ℝ^{n} → ℝ^{n} of the form F = Id + F_{(2)} + F_{(4)} + ... + F_{(2m)}, and also for all polynomial mappings F of the form F = Id + F_{(2)} − F_{(3)} + ... + (−1)^{m}F_{(m)}, where F_{(i)} are morphisms defined by homogeneous polynomials of degree i with non-negative coefficients.

2 Nilpotent Mappings

2.1. It is well-known that a matrix A ∈ M(n, ℂ) is nilpotent if and only if det(tE − A) ≡ t^{n}, where E is the identity matrix. As a consequence we obtain the following Lemma.

Lemma 1 A morphism N : ℂ^{n} → ℂ^{n} given by polynomials N_{i}(X) is nilpotent if and only if F̃ = (Id − T · N) × Id : ℂ^{n} × ℂ^{1} → ℂ^{n} × ℂ^{1}, given by the polynomials

\begin{aligned}Ñ_{i}(X_{1}, ..., X_{n}, T) & = & X_{i} − T N_{i}(X_{1}, ..., X_{n}) for i = 1, ... , n, \\ F̃_{n+1}(X_{1}, ..., X_{n}, T) & = & T , \end{aligned}

has F̃′(X, T) ≡ 1.

If we apply lemma 1 to the restriction of F̃ = Id − T · N to the hypersurface ℂ^{n} = { T = 1 } we obtain that for the nilpotent mapping N the corresponding morphism F = Id−N : ℂ^{n} → ℂ^{n} has F′(X) ≡ 1. Besides, if there exist d distinct fixed points of N, then for F = Id − N the preimage of the origin consists of d distinct points, and hence degree of F = Id − N is not less than d. In particular, if a nilpotent mapping N : ℂ^{n} → ℂ^{n} possesses d > 1 fixed points, then F = Id − N : ℂ^{n} → ℂ^{n} is not invertible. 2.2. Theorem 2 is a consequence of the following theorem.

Theorem 8 Any morphism F : ℂ^{n} → ℂ^{n} of degree deg F = d with F′(X) ≡ 1 is stably equivalent to \overline{F} of the form

\overline{F} = Id − N,

where N is a nilpotent morphism possessing d fixed points.

Conversely, if a nilpotent mapping N : ℂ^{n} → ℂ^{n} possesses d distinct fixed points, then \overline{F} = Id − N is of degree not less than d.

Proof. Let F : ℂ^{n} → ℂ^{n} be a morphism of degree deg F = d with F′(X) ≡ 1. Performing a translation and linear transformations, if necessary, we can assume that F (0) = 0, the number of the preimages of the origin #{F^{−1}(0)} = deg F, and F is given by polynomials

F_{i}(X_{1}, ..., X_{n}) = X_{i} − F_{(≥2)i}(X_{1}, ..., X_{n}),  (4)

where F_{(≥2)i}(X_{1}, ..., X_{n}) are polynomials containing no monomials of degree zero and one. Consider the blow up F_{t} : ℂ^{n+1} → ℂ^{n+1} of F. The morphism F_{t} is defined by polynomials

\begin{aligned}Y_{i} & = & F̃_{i}(X_{1}, ..., X_{n}, T) & = & X_{i} − ∑_{j=2}^{deg F_{i}} T^{j−1}F_{(j)i}(X_{1}, ..., X_{n}), i = 1, ..., n, \\ T̃ & = & F̃_{n+1}(X_{1}, ..., X_{n}, T) & = & T. \end{aligned}

Let us show that F_{t} is stably equivalent to G̃ = (Id−T ·G)×Id : ℂ^{m} ×ℂ^{1} → ℂ^{m} ×ℂ^{1} given by Ỹ_{i} = G̃_{i}(X_{1}, ..., X_{m}, T) = X_{i} − T G_{i}(X_{1}, ..., X_{m}) for i = 1, ..., m, and G̃_{m+1}(X_{1}, ..., X_{n}, T) = T, and of degree deg G̃ = deg F, and the number of the preimage of the point with coordinates (0, ..., 0, 1) is #{G̃^{−1}((0, ..., 0, 1))} = deg F. For this we must get rid of all monomials of the form T^{k}X_{1}^{r_{1}} ⋯ X_{n}^{r_{n}} with k > 1 and then apply Lemma 1 to G : ℂ^{m} → ℂ^{m} defined by G_{i}(X_{1}, ..., X_{m}), i = 1, ..., m. By Lemma 1, G is nilpotent and the number of the fixed points of G is equal to deg F.

Removing the monomial T^{k}X_{1}^{r_{1}} ⋯ X_{n}^{r_{n}} with k > 1 can be performed by adding new variables (passing to stably equivalent morphisms) and performing linear and triangular transformations. In fact, let for some i a polynomial Ỹ_{i} be of the form

Ỹ_{i} = X̃_{i} − ∑_{j=1}^{k_{i}} T^{j}F_{(j)i}(X̃_{1}, ..., X̃_{n})

(here F_{(j)i}(X̃_{1}, ..., X̃_{n}) is not necessary homogeneous). We add a new variable Ỹ_{n+1} = X̃_{n+1} and perform a triangular transformation in the preimage:

\begin{aligned}X_{i} & = & X̃_{i}, & i = 1, ..., n \\ X_{n+1} & = & X̃_{n+1} + T F_{(k_{i})i}(X̃_{1}, ..., X̃_{n}). \end{aligned}

Next we perform a triangular transformation in the image:

\begin{matrix}Y_{l} = & Ỹ_{l}, & l = 1, ..., i − 1, i + 1, ..., n + 1 \\ Y_{i} = & Ỹ_{i} − T^{k_{i}−1}Ỹ_{n+1}. \end{matrix}

It is easy to see that in the coordinates X_{1}, ..., X_{n+1}, T and Y_{1}, ..., Y_{n+1}, T the morphism F_{t} × Id is given by equations:

Y_{s} = X_{s} − ∑_{j=1}^{k_{s}} T^{j}F_{(j)s}(X_{1}, ..., X_{n}),  s = 1, ..., i − 1, i + 1, ..., n
Y_{i} = X_{i} − ∑_{j=1}^{k_{i}−1} T^{j}F_{(j)l}(X_{1}, ..., X_{n}) − T^{k_{i}−1}X_{n+1},
Y_{n+1} = X_{n+1} − T F_{(k_{i})i}(X_{1}, ..., X_{n}).

Thus, we removed the summand of F̃_{i}(X_{1}, ..., X_{n}, T) containing the variable T is contained with the maximal multiplicity k_{i}. Obviously, after a finite number of similar steps we obtain a morphism stably equivalent to F_{t} and satisfying the required conditions. 2.3. Let F : ℂ^{n} → ℂ^{n} be of the form F = Id − F_{(k)}, i.e. F is defined by

F_{i}(X) = X_{i} − F_{(k)i}(X_{1}, ..., X_{n}), i = 1, ..., n,

where F_{(k)i}(X_{1}, ..., X_{n}) are homogeneous forms of degree k. If F′(X) ≡ 1, then F_{(k)} is nilpotent. In fact, by Theorem 2, the Jacobian F′_{t}(X, T) of the blow up F_{t} is equal to

F′_{t}(X, T) = F′(T X_{1}, ..., T X_{n}) = det(E − T^{k−1}JF_{(k)}) ≡ 1.

Therefore, the Jacobian of F̂_{t} which is defined by polynomials

\begin{aligned}Y_{i} & = & X_{i} − T F_{(k)i}(X_{1}, ..., X_{n}), & i = 1, ..., n, \\ T & = & T, \end{aligned}

is also equal to 1. Hence, by Lemma 1, F_{(k)} is nilpotent.

Proposition 2 The conjecture JN(n) holds for nilpotent morphisms N_{(k)} : ℂ^{n} → ℂ^{n} defined by homogeneous forms of degree k.

Proof. The origin, i.e. the point o = (0, ..., 0), is a fixed point of N = N_{(k)}. Let x = (x_{1}, ..., x_{n}) ≠ o be another fixed point of N_{(k)}. This contradicts the nilpotency of N, since the vector \overline{x} = (x_{1}, ..., x_{n}) is an eigenvector (corresponding to a non-zero eigenvalue (equals to k)) of the linear mapping JN_{(k)}(X) at the point X = x. In fact, by Euler’s formula

JN_{(k)}(x)(\overline{x}) = (∑_{j=1}^{n} x_{j} \frac{∂N_{1}}{∂X_{j}} (x), ..., ∑_{j=1}^{n} x_{j} \frac{∂N_{n}}{∂X_{j}} (x))
= (kN_{1}(x), ..., kN_{n}(x)) =
= (kx_{1}, ..., kx_{n}) = k · \overline{x},

where N_{i}(X) are polynomials defining N = N_{(k)}.

Theorem 6 is an easy consequence of this theorem. For let F be of polynomial degree deg_{p} F = 2. After a linear changes of variables, we can assume that F is of the form F = Id−F_{(2)} and the preimage F^{−1}(0) consists of deg F distinct points. Hence, the nilpotent morphism F_{(2)} possesses deg F distinct fixed points. Therefore, deg F = 1, i.e. F is invertible.

As a consequence of above considerations, Lemma 1, Theorem 6, and the poof of Theorem 8 we obtain the following theorem.

Theorem 9 The conjecture JN(n) holds for N : ℂ^{n} → ℂ^{n} of polynomial degree deg_{p} N ≤ 2.

Proposition 2 may be generalized as follows.

Proposition 3 The conjecture JN(n) holds for N : ℂ^{n} → ℂ^{n} of the form N = N_{(k_{1})} + N_{(k_{2})}, where N_{(k_{i})} are morphisms defined by homogeneous forms of degree k_{i} > 0.

Proof. The origin is a fixed point of N. Let x = (x_{1}, ..., x_{n}) ≠ o be another fixed point of N. As in the proof of Proposition 2 this contradicts the nilpotency of N, since the vector \overline{x} = (x_{1}, ..., x_{n}) is an eigenvector (corresponding to a non-zero eigenvalue) of the linear mapping JN(X) at the point X = λx = (λx_{1}, ..., λx_{n}), where

λ = (\frac{k_{1}}{k_{2}} )^{\frac{1}{k_{2}−k_{1}}} .

In fact,

JN(λx)(\overline{x}) = (∑_{j=1}^{n} x_{j} \frac{∂N_{(k_{1})1}}{∂X_{j}} (λx), ..., ∑_{j=1}^{n} x_{j} \frac{∂N_{(k_{1})n}}{∂X_{j}} (λx))+
+ (∑_{j=1}^{n} x_{j} \frac{∂N_{(k_{2})1}}{∂X_{j}} (λx), ..., ∑_{j=1}^{n} x_{j} \frac{∂N_{(k_{2})n}}{∂X_{j}} (λx)) =
= (k_{1}λ^{k_{1}−1}N_{(k_{1})1}(x), ..., k_{1}λ^{k_{1}−1}N_{(k_{1})n}(x))+
+ (k_{2}λ^{k_{2}−1}N_{(k_{2})1}(x), ..., k_{2}λ^{k_{2}−1}N_{(k_{2})n}(x)) =
= k_{1} (\frac{k_{1}}{k_{2}} )^{\frac{k_{1}−1}{k_{2}−k_{1}}} · N_{(k_{1})}(x) + k_{2} (\frac{k_{1}}{k_{2}} )^{\frac{k_{2}−1}{k_{2}−k_{1}}} · N_{(k_{2})}(x) =

\begin{matrix}= & \frac{k \frac{k_{2}−1}{k_{2}−k_{1}} ; 1}{k \frac{k_{1}−1}{k_{2}−k_{1}} ; 2} · N_{(k_{1})}(x) + \frac{k \frac{k_{2}−1}{k_{2}−k_{1}} ; 1}{k \frac{k_{1}−1}{k_{2}−k_{1}} ; 2} · N_{(k_{2})}(x) \\ = & \frac{k \frac{k_{2}−1}{k_{2}−k_{1}} ; 1}{k \frac{k_{1}−1}{k_{2}−k_{1}} ; 2} · (N_{(k_{1})}(x) + N_{(k_{2})}(x)) \\ = & \frac{k \frac{k_{2}−1}{k_{2}−k_{1}} ; 1}{k \frac{k_{1}−1}{k_{2}−k_{1}} ; 2} · \overline{x}. \end{matrix}

2.4. The following result is a consequence of Theorem 3 and the proof of Theorem 8

Theorem 10 If each nilpotent morphism N : ℂ^{n} → ℂ^{n} of polynomial degree deg_{p} N = 3 has a unique fixed point, then JN(n) holds.

In the case deg_{p} N = 3 the morphism N is of the form

N = N_{(1)} + N_{(2)} + N_{(3)},

where N_{(i)} is a homogeneous form of degree i, and Proposition 3 can be reformulated as follows:

The conjecture JN(n) holds for N : ℂ^{n} → ℂ^{n} of the form N = N_{(1)} + N_{(3)} and N = N_{(2)} + N_{(3)}.

It follows from [6] that:

The conjecture JN(3) holds for N : ℂ^{3} → ℂ^{3} of polynomial degree deg_{p} N ≤ 3. 2.5. The following proposition is a consequence of Theorem 7.

Proposition 4 Let F : ℝ^{n} → ℝ^{n} be a polynomial mapping of the form F = Id + N, where N is a positive nilpotent morphism. Then F is invertible.

Proof. Without loss of generality, we can assume that N(0) = 0. Consider a morphism \overline{F} : ℝ^{n+1} → ℝ^{n+1} defined by \overline{F}_{i} = X_{i} − T N_{i}(X), i = 1, ..., n, and \overline{F}_{n+1} = T. By Theorem 7, \overline{F} is invertible. The restriction of \overline{F} to the hyperplane T = −1 coincides with F. Therefore, F is invertible.

In particular, it follows from this that:

The conjecture JN(n) holds for N : ℝ^{n} → ℝ^{n} if N is either a positive or a negative nilpotent morphism.

It follows from Theorem 7 (if we use a blow up) that:

The conjecture JN(n) holds for the nilpotent morphisms N : ℝ^{n} → ℝ^{n} of the form

N = ±(N_{(1)} − N_{(2)} + ... + (−1)^{m+1}N_{(m)}),

where N_{(i)} are defined by positive homogeneous forms of degree i. 2.6. Let r = rk JN(X) be the rank of a nilpotent morphism N : ℂ^{n} → ℂ^{n}. In [2] it was shown that if r = 1, then F = Id − N is invertible:

The conjecture JN(n) holds for nilpotent morphisms N : ℂ^{n} → ℂ^{n} of rank r = 1. In particular, the conjecture JN(2) is true.

References

[1] H. Appelgate, H. Onishi, The Jacobian Conjecture in two variables // J.P.A.A., 37 (1985), 215-227.

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Steklov Mathematical Institute victor@olya.ips.ras.ru kulikov@mpim-bonn.mpg.de