Some classes satisfying the 2-dimensional Jacobian conjecture and a proof of the complex conjecture until degree 104

TL;DR

This paper constructs specific polynomial maps to explore the Jacobian conjecture in two variables, proving it holds for degrees up to 104 in the complex case, thus extending previous known bounds.

Contribution

It introduces new classes of polynomial maps satisfying the 2D Jacobian conjecture and improves the degree bound for the complex case from 100 to 104 using Newton polygon techniques.

Findings

Identified classes of polynomial maps satisfying the Jacobian conjecture.

Proved the conjecture holds for degrees up to 104 in the complex case.

Extended the known degree bound from 100 to 104 since 1983.

Abstract

We construct a non-proper set of two variables polynomial maps and study the nowhere vanishing Jacobian condition of the Jacobian conjecture for this set. We obtain some classes of polynomial maps satisfying the 2-dimensional Jacobian conjecture for both real and complex cases. In addition, by Newton polygon technique, we prove that the complex conjecture is true until degree 104, improving Moh boundary (degree 100) since 1983.

Equations

$$\det (JF(x,y))=\mathrm{constant}\ne 0,\forall (x,y)\in {\mathbb{K}}^2,$$

$$S_F=\{a\in {\mathbb{K}}^2:\text{ there is a sequence }{z}_k\to \infty ,F(z_k)\to a\}.$$

$$t=xy-1,h=t(xt+1),f=(xt+1{)}^2(t^2+y),$$

$$q=-t^2-6th(h+1)-u(f,h)$$

$$\det (J(p,q))=t^2+f^2+(t+f(13+15h){)}^2$$

$$F(x,y)=H(u_0(x,y),u_1(x,y),\dots ,u_n(x,y)),$$

$$F(x,y)=(H_1(u_0,u_1,\dots ,u_n),H_2(u_0,u_1,\dots ,u_n)).$$

$$\left(\frac{\partial u_0}{\partial x}(z),\frac{\partial u_1}{\partial x}(z),\dots ,\frac{\partial u_n}{\partial x}(z)\right),$$

$$\left(\frac{\partial u_0}{\partial y}(z),\frac{\partial u_1}{\partial y}(z),\dots ,\frac{\partial u_n}{\partial y}(z)\right)$$

$$\left(\frac{\partial u_0}{\partial x}(z),\frac{\partial u_1}{\partial x}(z),\dots ,\frac{\partial u_n}{\partial x}(z)\right),$$

$$\left(\frac{\partial u_0}{\partial y}(z),\frac{\partial u_1}{\partial y}(z),\dots ,\frac{\partial u_n}{\partial y}(z)\right)$$

$$F(x,y)=(x-2(xy+1)-y(xy+1{)}^2,-1-y(xy+1)).$$

$$u_0=x,u_1=xy+1,u_2=y(xy+1),u_3=y(xy+1{)}^2,$$

$$H:{\mathbb{C}}^4\to {\mathbb{C}}^2,H(u_0,u_1,u_2,u_3)=(u_0-2u_1-u_3,-1-u_2)$$

$$\begin{array}{cccccc}\left(\frac{\partial u_0}{\partial x}(z),\frac{\partial u_1}{\partial x}(z),\frac{\partial u_2}{\partial x}(z),\frac{\partial u_3}{\partial x}(z)\right)& =& (1,& y,& y^2,& 2x{y}^3+2y^2),\\ \left(\frac{\partial u_0}{\partial y}(z),\frac{\partial u_1}{\partial y}(z),\frac{\partial u_2}{\partial y}(z),\frac{\partial u_3}{\partial y}(z)\right)& =& (0,& x,& 2xy+1,& 3x^2{y}^2+4xy+1).\end{array}$$

$$F_1(x,y)=H_1(u_0(x,y),u_1(x,y),\dots ,u_n(x,y)),$$

$$F_2(x,y)=H_2(u_0(x,y),u_1(x,y),\dots ,u_n(x,y)).$$

$$JF(x,y)=\left(\begin{array}{cc}\frac{\partial F_1}{\partial x}& \frac{\partial F_1}{\partial y}\\ \frac{\partial F_2}{\partial x}& \frac{\partial F_2}{\partial y}\end{array}\right)=\left(\begin{array}{cccc}\frac{\partial H_1}{\partial u_0}& \frac{\partial H_1}{\partial u_1}& \cdots & \frac{\partial H_1}{\partial u_n}\\ \frac{\partial H_2}{\partial u_0}& \frac{\partial H_2}{\partial u_1}& \cdots & \frac{\partial H_2}{\partial u_n}\end{array}\right)\left(\begin{array}{cc}\frac{\partial u_0}{\partial x}& \frac{\partial u_0}{\partial y}\\ \frac{\partial u_1}{\partial x}& \frac{\partial u_1}{\partial y}\\ ⋮& ⋮\\ \frac{\partial u_n}{\partial x}& \frac{\partial u_n}{\partial y}\end{array}\right).$$

$$\left(\frac{\partial u_0}{\partial x}(z),\frac{\partial u_1}{\partial x}(z),\dots ,\frac{\partial u_n}{\partial x}(z)\right),$$

$$\left(\frac{\partial u_0}{\partial y}(z),\frac{\partial u_1}{\partial y}(z),\dots ,\frac{\partial u_n}{\partial y}(z)\right)$$

$$t=xy-1,h=t(xt+1),f=(xt+1{)}^2(t^2+y),$$

$$q=-t^2-6th(h+1)-u(f,h)$$

$$\det (J(p,q))=t^2+f^2+(t+f(13+15h){)}^2$$

$$f(x,y)=(xt(x,y)+1{)}^2(t^2(x,y)+y).$$

$$h(x,y)-t(x,y)=-f(x,y)=-f+3h^2+2h=0.$$

$$\det J(p,-t^2-6th(h+1)-u(f,h))=t^2+(t+(f(13+15h){)}^2+f^2.$$

$$p(t,h,f)=f+h,q(t,h,f)=-t^2-6th(h+1)-u(f,h).$$

$$(p,q):{\mathbb{K}}^2\to {\mathbb{K}}^2,(p,q)(x,y):=H(t(x,y),h(x,y),f(x,y)).$$

$$u(f,h)=170fh+91h^2+195f{h}^2+69h^3+75f{h}^3+\frac{75}{4}{h}^4,$$

$$q=-\frac{(h^3+h^2{)}^2}{f^2}-\frac{2h^2(h^3+h^2)}{f}+4h^3+\frac{3h^2}{2}-5hf,$$

Full text

1. Introduction

Let $F=(F_{1},F_{2}):\mathbb{K}^{2}\to\mathbb{K}^{2}$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, be a polynomial map satisfying the condition:

[TABLE]

where $JF(x,y)$ is the Jacobian matrix of $F$ at $(x,y)$. The famous Jacobian conjecture, stated by O. H. Keller [12] in 1939, saying that any polynomial map $F:\mathbb{K}^{2}\to\mathbb{K}^{2}$ with the condition (1.0) is an automorphism, is still open today when $\mathbb{K}=\mathbb{C}$. In the real case, the conjecture was solved negatively by Pinchuk [20] in 1994. In this paper, following the book [8] we call the condition (1.0) Jacobian condition.

If $F$ has a global inverse, then its inverse, being continous, maps compact sets into compact sets, in other words $F$ is proper. The smallest set $S_{F}$ such that the map $F:\mathbb{K}^{2}\,\setminus\,F^{-1}(S_{F})\to\mathbb{K}^{2}\,\setminus\,S_{F}$ is proper is called the asymptotic set of $F$. The Jacobian conjecture reduces to show that the asymptotic set of a polynomial map satisfying the Jacobian condition is empty. For polynomial maps, this set can be expressed as the set of points in the target space such that there is a sequence $\{z_{k}\}\subset\mathbb{K}^{2}$ tending to infinity but its image $F(z_{k})$ converges to a finite point:

[TABLE]

This paper is inspired by the counterexamples of Pinchuk [20]: given $(x,y)\in\mathbb{R}^{2}$, denote

[TABLE]

then the Pinchuk maps $F=(p,q)$ are the ones with $p=f+h$ and $q$ varies for different maps $(p,q)$ but $q$ always has the form

[TABLE]

where $u$ is an auxiliary polynomial in $f$ and $h$ and is chosen such that

[TABLE]

([20, Lemma 2.2]). Then ${\rm det}(J(p,q)(x,y))>0$ for every $(x,y)\in\mathbb{R}^{2}$ since if $t=0$ then $f=y=\frac{1}{x}\neq 0$. However, $(p,q)$ is not automorphism. Our first observation is: by changing variables $t:=t(x,y)$, $h:=h(x,y)$ and $f:=f(x,y)$, we can write the Pinchuk maps $(p,q)$ as polynomial maps of new variables $t,h$ and $f$. The sequence $\{z_{k}\}=\{(k,1/k)\}\subset\mathbb{R}^{2}$ tends to infinity but $t(z_{k})=h(z_{k})=0$ and $f(z_{k})=1/k$. Then the Pinchuk maps $(p,q)$, being polynomial maps of new variables $t,h$ and $f$, satisfy that $(p(z_{k}),q(z_{k}))$ tends to a finite point and therefore they are non-proper and hence, is not automorphism. We call the set of such new variables $\{t,h,f\}$ a set of non-proper variables of the map $F=(p,q)$. By studying the Jacobian condition for polynomial maps of these new variables and classifying them, we obtain some classes that satisfy the Jacobian conjecture for both real and complex cases.

Structure and Results of the paper

The paper has two parts.

The first part, presented in Section 2, concerns the construction of non-proper variables for both real and complex cases and a classification of non-proper maps written under these non-proper variables. We obtain some explicit classes satisfying the 2-dimensional Jacobian conjecture for both real and complex cases (Corollaries 2.19, 2.21 and 2.22). These results are classified in Remark 2.24. We also provide a possible counter-example for the conjecture (if there exists) in Corollary 2.25.

The second part, presented in Section 3, concerns a particular result to the complex conjecture for degree until 104 (Theorem 3.6). We prove that the complex conjecture is true for degree until 104 using known results studied by Abhyankar [2], Nagata [17, 18], Appelgate and Onishi [3], Magnus [13], Nakai and Baba [16] and Żoła̧dek [24]. It is known that the complex conjecture is true for degree until 100 (Moh [14], 1983). There are some works increasing the degree 100 of Moh, for instance [10], but we emphasize that our proof is a very simple one. There may be a connection between the two parts of the paper for higher degree cases that we intend to study in our next step (Remark 3.8).

2. Non-proper variables sets of a polynomial map

2.1. Construction and examples

We consider $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and $F=(F_{1},F_{2}):\mathbb{K}^{2}\to\mathbb{K}^{2}$ a polynomial map of two variables $x$ and $y$. With $z\in\mathbb{K}^{2}$, we denote by $\|z\|$ its Euclidean norm. Let us recall firstly that the polynomial map $F$ is non proper if there is a sequence $\{z_{k}=(x_{k},y_{k})\}_{k\in\mathbb{N}}\subset\mathbb{K}^{2}$ such that $\lim_{k\to\infty}\|z_{k}\|=+\infty$ and $\lim_{k\to\infty}F(z_{k})=(c_{1},c_{2})$, for some finite $c_{1},c_{2}\in\mathbb{K}$. The following remark is important for the definition of non-proper variables sets that we will introduce in Definition 2.2.

Remark 2.1**.**

$F$* is non-proper if and only if it can be written as a polynomial map of some variables $u_{0},u_{1},\dots,u_{n}$ ($\,n\geq 1$), where $u_{0},u_{1},\dots,u_{n}$ are polynomial functions of two variables $x$ and $y$ such that there is a sequence $\{z_{k}=(x_{k},y_{k})\}\subset\mathbb{K}^{2}$ with $\lim_{k\to\infty}\|z_{k}\|=+\infty$ but $\lim_{k\to\infty}u_{i}(z_{k})=\alpha_{i}\in\mathbb{K}$ for all $\alpha_{i}$ are finite numbers, i.e.*

[TABLE]

where $H=(H_{1},H_{2}):\mathbb{K}^{n+1}\to\mathbb{K}^{2}$ is a polynomial map of variables $u_{0},u_{1},\dots,u_{n}$.

In fact, if $F=(F_{1},F_{2})$ is non proper, we can take $u_{0}:=F_{1}$ and $u_{1}:=F_{2}$ and the inverse is obvious. We define:

Definition 2.2** (non-proper variables sets).**

Let $F=(F_{1},F_{2}):\mathbb{K}^{2}\to\mathbb{K}^{2}$ be a polynomial map. A set of polynomial functions $u_{0},u_{1},\dots,u_{n}:\mathbb{K}^{2}\to\mathbb{K}\,(n\geq 1)$ is called a set of non-proper variables of $F$ if there exists

1. (1)

a sequence $\{z_{k}\}_{k\in\mathbb{N}}\subset\mathbb{K}^{2}$ such that $\lim_{k\to\infty}\|z_{k}\|=+\infty$ and $\lim_{k\to\infty}u_{i}(x_{k},y_{k})$ is a finite number $\alpha_{i}\in\mathbb{K}$ for all $i=0,\dots,n$; 2. (2)

a polynomial map $H=(H_{1},H_{2}):\mathbb{K}^{n+1}\to\mathbb{K}^{2}$ such that

[TABLE]

Remark 2.3**.**

A polynomial map $F:\mathbb{K}^{2}\to\mathbb{K}^{2}$ is non-proper if and only if it possesses at least a set of non-proper variables $\{u_{0},u_{1},\dots,u_{n}\}$. **

Definition 2.4** (Dependent/Independent non-proper variables sets).**

Let $F:\mathbb{K}^{2}\to\mathbb{K}^{2}$ be a non-proper polynomial map and $\{u_{0},u_{1},\dots,u_{n}\}$ is a set of non-proper variables of $F$. We say that:

1. (1)

$\{u_{0},u_{1},\dots,u_{n}\}$ is dependent if there exists $z=(x,y)\in\mathbb{K}^{2}$ such that the two vectors

[TABLE]

[TABLE]

are linearly dependent in $\mathbb{K}^{n+1}$; 2. (2)

otherwise, $\{u_{0},u_{1},\dots,u_{n}\}$ is independent if for any $z=(x,y)\in\mathbb{K}^{2}$, the two vectors

[TABLE]

[TABLE]

are linearly independent in $\mathbb{K}^{n+1}$.

Example 2.5** ([19, Section 6]).**

Let $F:\mathbb{C}^{2}\to\mathbb{C}^{2}$ be the polynomial map defined by:

[TABLE]

Put:

[TABLE]

then $\{u_{0},u_{1},u_{2},u_{3}\}$ is a set of non-proper variables of $F$ since any sequence $\{(1/k,-k+c)\}$, with $c\in\mathbb{K}$, satisfies the first condition in Definition 2.2. The second condition is verified by taking

[TABLE]

and then $F=H(u_{0},u_{1},u_{2},u_{3})$. Consequently, $F$ is non-proper.

Now we show that $\{u_{0},u_{1},u_{2},u_{3}\}$ is independent. In fact, for any $z=(x,y)$, we have

[TABLE]

We see that there is no point $z\in\mathbb{C}^{2}$ such that the second partial derivative at $z$ is the zero vector in $\mathbb{C}^{4}$. Then the two partial derivatives above are linearly independent vectors at any point $z\in\mathbb{C}^{2}$. By Definition 2.4, the set of non-proper variables $\{u_{0},u_{1},u_{2},u_{3}\}$ is independent.

Let us remark that if we consider the above map as a real one $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$, i.e. the same map $F$ but with real variables, then $\{u_{0},u_{1},u_{2},u_{3}\}$ is also an independent set of non-proper variables.**

Proposition 2.6**.**

Let $F=(F_{1},F_{2}):\mathbb{K}^{2}\to\mathbb{K}^{2}$ be a non-proper polynomial map and $\{u_{0},u_{1},\dots,u_{n}\}$ be a set of non-proper variables of $F$. If $\{u_{0},u_{1},\ldots,u_{n}\}$ is dependent then $F$ does not satisfy the Jacobian condition.

Proof.

By Definition 2.2, there exists a polynomial map $H=(H_{1},H_{2}):\mathbb{K}^{n+1}\to\mathbb{K}^{2}$ such that

[TABLE]

[TABLE]

We have

[TABLE]

Since $u_{0},u_{1},\ldots,u_{n}$ are dependent, then by Definition 2.4, there exists $z=(x,y)\in\mathbb{K}^{2}$ such that the two vectors

[TABLE]

[TABLE]

are linearly dependent. It follows that ${\rm det}(JF(z))=0$. ∎

Corollary 2.7**.**

For any set of dependent non-proper variables $u_{0},u_{1},\ldots,u_{n}:\mathbb{K}^{2}\to\mathbb{K}$ and for any polynomial map $H=(H_{1},H_{2}):\mathbb{K}^{n+1}\to\mathbb{K}^{2}$, then $F:=H(u_{0},u_{1},\ldots,u_{n})$ is a polynomial map $\mathbb{K}^{2}\to\mathbb{K}^{2}$ satisfying the Jacobian conjecture.

Corollary 2.8**.**

Let $F=(F_{1},F_{2}):\mathbb{K}^{2}\to\mathbb{K}^{2}$ be a non-proper polynomial map and $\{u_{0},u_{1},\dots,u_{n}\}$ be a set of non-proper variables of $F$. If $F$ satisfies the Jacobian condition then $\{u_{0},u_{1},\ldots,u_{n}\}$ is independent.

In the following example we will show that the functions $t,h,f$ defined in [20] and also in [9] form a set of non-proper variables for the Pinchuk maps and it is independent. Let us remark that we will consider both of real and complex cases, i.e. we consider the polynomials $t,h,f:\mathbb{K}^{2}\to\mathbb{K}$, with $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. In the complex case, it shows that the inverse of Corollary 2.8 is not true.

Example 2.9**.**

Let us recall firstly the Pinchuk map defined in [20]: given $(x,y)\in\mathbb{R}^{2}$, denote

[TABLE]

then the Pinchuk maps $(p,q)$ are the ones with $p=f+h$ and $q$ varies for different maps $(p,q)$ but $q$ always has the form

[TABLE]

where $u$ is an auxiliary polynomial in $f$ and $h$ is chosen such that

[TABLE]

([20, Lemma 2.2]).

As remarked in [20], we have $\dfrac{h(x,y)+1}{x}=t^{2}(x,y)+y$. Thus, we may write:

[TABLE]

Considering the polynomials $t,h,f$ as both real and complex polynomials and the Pinchuk maps $(p,q)$ as both real and comlex polynomial mappings, i.e. we consider $t,h,f:\mathbb{K}^{2}\to\mathbb{K}$ and $(p,q):\mathbb{K}^{2}\to\mathbb{K}^{2}$, with $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, we have the following affirmations:

1. (1)

$\{t,h,f\}$ is a set of non-proper variables in both cases $\mathbb{K}=\mathbb{R}$ and $\mathbb{C}$.

In fact, let $z_{k}:=(k,1/k)$, then $t(z_{k})=h(z_{k})=0$ and $f(z_{k})=1/k$. 2. (2)

$\{t,h,f\}$ is a set of independent non-proper variables (Definition 2.4) in both cases $\mathbb{K}=\mathbb{R}$ and $\mathbb{C}$.

In fact, from [20, Lemma 2.1], it is enough to show that there is no $(x,y)\in\mathbb{K}^{2}$ such that

[TABLE]

Thus, the proof follows by a direct computation. 3. (3)

There exists a polynomial map $H:\mathbb{K}^{3}\to\mathbb{K}^{2}$ such that $(p,q)(x,y)=H(t,h,f).$

In fact, as in [20, page 3], set $p=f+h$, then by the Lemma 2.2 in [20], there exists a polynomial $u(f,h)$ such that

[TABLE]

We consider $H=(p,q):\mathbb{K}^{3}\to\mathbb{K}^{2}$ the polynomial map defined by

[TABLE]

The polynomials $p,q$ yield the polynomial map $H:\mathbb{K}^{3}\to\mathbb{K}^{2}$, where $H(t,h,f)=(p(t,h,f),q(t,h,f))$. Then the Pinchuk maps are defined in [20] by:

[TABLE]

Remark 2.10**.**

As the remark at the end of the paper [20], the Pinchuk map constructed in the proof of [20, Lemma 2.2] has degree 40, where ${\rm deg}(p)=10$ and ${\rm deg}(q)=40$, but one can reduce ${\rm deg}(q)$ to 35 by a suitable choice of the auxiliary polynomial $u(f,h)$. In fact, the polynomial $q$ varies for different Pinchuk map, but always has the form $q=-t^{2}-6th(h+1)-u(f,h)$. In [8] (page 241), van den Essen choose

[TABLE]

and in this case the degree of Pinchuk map is 25. This is also the one that Andrew Campbell studies in the series of his papers [5, 6, 7]. Of course, this Pinchuk map also satisfies all the affirmations in the Example 2.9. **

Example 2.11**.**

In [9], basing on Pinchuk’s examples [20], the author provides new counterexamples to the real conjecture but with the possibility to reduce the degree of $q$ to 15: In [9, Corollary 2.6], they prove that if taking $p=f+h$ and

[TABLE]

where $h$ and $f$ are the same polynomials defined in Pinchuk’s examples [20], then $(p,q)$ is also a non-injective polynomial local diffeomorphism. In this case ${\rm deg}\,p=10$ and ${\rm deg}\,q=15$.

With the same polynomial $t$ defined in Pinchuk maps [20], one has

[TABLE]

(see the proof of [9, Proposition 2.3]). It follows:

[TABLE]

Then, in this case $q$ is also a polynomial of non-proper variables $t,h,f$. In other words, the examples of “Pinchuk type” constructed in [9] are also polynomial maps of non-proper variables $t,h$ and $f$ satisfying all the affirmations in Example 2.9. **

2.2. A set of non-proper variables

In this section, we consider the following set of polynomial functions $u_{0},u_{1},\dots,u_{n}:\mathbb{K}^{2}\to\mathbb{K}\;(n\geq 1)$:

[TABLE]

The motivation for studying this set of variables is that any polynomial written under the variables $u_{0},u_{1},\ldots,u_{n}$:

[TABLE]

has Newton polygon containing two points (1, 0) and (0, 1) (see Definition of Newton polygon in [2], [17, 18], also in [8, page 93]). In the following we will prove first that for any triple of non-zero natural numbers $(n,r,s)$ and for any polynomial mapping $F:\mathbb{K}^{2}\to\mathbb{K}^{2}$ such that

[TABLE]

where $H=(H_{1},H_{2}):\mathbb{K}^{n+1}\to\mathbb{K}^{2}$, then $\{u_{0},u_{1},\dots,u_{n}\}$ is a set of non-proper variables of $F$ (Proposition 2.12 for complex case and Proposition 2.13 for real case). Then we investigate the independence (Definition 2.4) for this set: we consider all the possibilities of triples $(r,s,n)$, we divide them into three cases: $r=1$ (section 2.2.2); $(n=1,\,r\geq 2)$ (section 2.2.3); and $(n\geq 2,r\geq 2)$ (section 2.2.4). We obtain some classes satisfying the Jacobian conjecture for polynomial mappings $F:\mathbb{K}^{2}\to\mathbb{K}^{2}$, $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ (Corollary 2.21 and Corollary 2.22).

2.2.1. The set of variables (2.11) is a set of non-proper variables

Proposition 2.12**.**

For any fixed $n\geq 1$, the set of variables $u_{0},u_{1},\ldots,u_{n}:\mathbb{C}^{2}\to\mathbb{C}$ in (2.11) is a set of non-proper variables of any polynomial map

[TABLE]

where $H=(H_{1},H_{2}):\mathbb{C}^{n+1}\to\mathbb{C}^{2}$ is a polynomial map of variables $u_{0},u_{1},\dots,u_{n}$.

Proof.

With a fixed $n\geq 1$ and given $r,s\geq 1$, let us consider the sequence $\{(x_{k},y_{k})\}\subset\mathbb{C}^{2}$, with

[TABLE]

We will prove that $u_{i}(x_{k},y_{k})$ does not tend to infinity for any $i=0,1,\ldots,n$.

At first, let us remark that $u_{0}(x_{k},y_{k})=y_{k}$ tends to $\sqrt[s]{1/b}$ if $r=1$ and tends to 0 if $r>1$. Now for $i=1,\ldots,n$, we have

[TABLE]

then, by a direct computation, it follows:

[TABLE]

where

[TABLE]

For $i=1,\dots,n-1$, the sequence $u_{i}(x_{k},y_{k})$ tends to 0 since in this case $nl-i>0$ for any $l\geq 1$. Now we have

[TABLE]

then $u_{n}(x_{k},y_{k})$ tends to $(-n)$. ∎

Now if we replace $\mathbb{C}$ by $\mathbb{R}$ in Proposition 2.12, then the sequence $\{(x_{k},y_{k})\}$ chosen in the above proof can be used if $b>0$. Therefore we have:

Proposition 2.13**.**

If $b>0$, then the set of variables $u_{0},u_{1},\ldots,u_{n}:\mathbb{R}^{2}\to\mathbb{R}$ in (2.11) is a set of non-proper variables of any polynomial map $F(x,y)=H(u_{0},u_{,}\ldots,u_{n}):\mathbb{R}^{2}\to\mathbb{R}^{2}$ where $H=(H_{1},H_{2}):\mathbb{R}^{n+1}\to\mathbb{R}^{2}$ is a polynomial map of variables $u_{0},u_{1},\dots,u_{n}$.

Remark 2.14**.**

Proposition 2.12 and Proposition 2.13 are still true if we change the sign of all the terms in the non-proper variables (2.11), i.e. they are true for the set of variables:

[TABLE]

Remark 2.15**.**

Proposition 2.12 and Proposition 2.13 are still true if we exchange the roles of $x$ and $y$.**

Consider the variables $u_{0},u_{1},\dots,u_{n}$ in (2.11).

Corollary 2.16**.**

If $F:\mathbb{C}^{2}\to\mathbb{C}^{2}$ can be written as a polynomial map of variables $u_{0},u_{1},\dots,u_{n}$, i.e. $F(x,y)=H(u_{0},u_{1},\ldots,u_{n})$, where $H=(H_{1},H_{2}):\mathbb{C}^{n+1}\to\mathbb{C}^{2}$ is a polynomial map of variables $u_{0},u_{1},\dots,u_{n}$, then $F$ is non-proper.

Corollary 2.17**.**

If $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$ can be written as a polynomial map of variables $u_{0},u_{1},\dots,u_{n}$, i.e. $F(x,y)=H(u_{0},u_{1},\ldots,u_{n})$, where $H=(H_{1},H_{2}):\mathbb{R}^{n+1}\to\mathbb{R}^{2}$ is a polynomial map of variables $u_{0},u_{1},\dots,u_{n}$ in (2.11) with $b>0$, then $F$ is non-proper.

2.2.2. The set of non-proper variables (2.11) with $r=1$

We consider the non-proper variables $u_{0},u_{1},\dots,u_{n}:\mathbb{K}^{2}\to\mathbb{K}$ in (2.11) with $r=1$, i.e. the set

[TABLE]

Here we consider $b\neq 0$ if $\mathbb{K}=\mathbb{C}$ (cf. Proposition 2.12) and $b>0$ if $\mathbb{K}=\mathbb{R}$ (cf. Proposition 2.13).

Theorem 2.18**.**

For any fixed $n\geq 1$, the set $\{u_{0},u_{1},\ldots,u_{n}\}$ in (2.17) is a set of dependent non-proper variables.

Proof.

By Definition 2.4, we show that there exists $z\in\mathbb{K}^{2}$ such that $\dfrac{\partial u_{i}}{\partial x}(z)=0$, for all $i=0,1,\dots,n$. In fact, we have

[TABLE]

Then any point $z=(x_{0},\sqrt[s]{1/b})$, where $x_{0}\in\mathbb{K}$, is a solution of the system $\dfrac{\partial u_{i}}{\partial x}(z)=0$, for $i=0,1,\ldots,n$ (notice that we consider $b\neq 0$ if $\mathbb{K}=\mathbb{C}$ and $b>0$ if $\mathbb{K}=\mathbb{R}$). ∎

It follows from Proposition 2.6 and Theorem 2.18 that:

Corollary 2.19**.**

Any polynomial map $F(x,y)=H(u_{0},u_{,}\ldots,u_{n}):\mathbb{K}^{2}\to\mathbb{K}^{2}$, where $H=(H_{1},H_{2}):\mathbb{K}^{n+1}\to\mathbb{K}^{2}$ is a polynomial map of variables $u_{0},u_{1},\dots,u_{n}$ in (2.17) satisfies the Jacobian conjecture.

2.2.3. The set of non-proper variables (2.11) with $n=1$ and $r\geq 2$

Consider the case $n=1$, i.e. the case where the set (2.11) has only two variables:

[TABLE]

Theorem 2.20**.**

We have:

1. (1)

In the complex case $u_{0},u_{1}:\mathbb{C}^{2}\to\mathbb{C}$, the set of non-proper variables $\{u_{0},u_{1}\}$ is dependent with any $r\geq 2$, $s\geq 1$ and $b\neq 0$; 2. (2)

In the real case $u_{0},u_{1}:\mathbb{R}^{2}\to\mathbb{R}$, the set of non-proper variables $\{u_{0},u_{1}\}$ is dependent with any $r\geq 2$, $s\geq 1$ and $b>0$.

Proof.

We have ${\partial u_{1}}/{\partial x}=1-rbx^{r-1}y^{s}$. With the same argument as in the proof of Theorem 2.18, the equation ${\partial u_{1}}/{\partial x}=0$ always has a solution in the complex case. In the real case, when $r$ is odd, $s$ is even and $b>0$, the equation ${\partial u_{1}}/{\partial x}=0$ admits al least a solution. ∎

From Proposition 2.6, Corollary 2.19 and Theorem 2.20, we have two explicit classes in which the two-dimensional Jacobian conjecture is true, for complex and real cases, respectively:

Corollary 2.21**.**

Let $F=(F_{1},F_{2}):\mathbb{C}^{2}\to\mathbb{C}^{2}$ be a polynomial map. If $F_{1}$ and $F_{2}$ can be written as a finite sum:

[TABLE]

where

[TABLE]

then $F$ satisfies the Jacobian conjecture.

Corollary 2.22**.**

Let $F=(F_{1},F_{2}):\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a polynomial map. If $F_{1}$ and $F_{2}$ can be written as a finite sum:

[TABLE]

where

[TABLE]

then $F$ satisfies the Jacobian conjecture.

2.2.4. The set of non-proper variables (2.11) with $n\geq 2$ and $r\geq 2$

We consider the set of non-proper variables $u_{0},u_{1},\dots,u_{n}$ in (2.11) with $r\geq 2$, i.e. the set

[TABLE]

Here we consider $b\neq 0$ if $\mathbb{K}=\mathbb{C}$ (cf. Propostion 2.12) and $b>0$ if $\mathbb{K}=\mathbb{R}$ (cf. Proposition 2.13).

Proposition 2.23**.**

*For a fixed natural number $n\geq 2$, the set of variables

$u_{0},u_{1},\ldots,u_{n}$ in (2.22) is a set of independent non-proper variables.*

Proof.

For any $z=(x,y)\in\mathbb{K}^{2}$, we have:

[TABLE]

[TABLE]

By Definition 2.4, we need to show that the two above vectors are linearly independent for any $z\in\mathbb{K}^{2}$. It is enough to show that there is no $z\in\mathbb{K}^{2}$ such that $\dfrac{\partial u_{1}}{\partial x}(z)=\dfrac{\partial u_{2}}{\partial x}(z)=0$. We have

[TABLE]

Let us suppose that $\dfrac{\partial u_{1}}{\partial x}(z)=\dfrac{\partial u_{2}}{\partial x}(z)=0$. The first equation implies $rbx^{r-1}y^{s}=1$, then $y\neq 0$. Since $r>1$, we have also $x\neq 0$. Therefore, the second equation implies $b\sqrt{r}x^{r-1}y^{s}=1$. From the two last equations, we have $rbx^{r-1}y^{s}=\sqrt{r}bx^{r-1}y^{s}$, that implies $r=1$, which is a contradiction because $r>1$. ∎

Remark 2.24**.**

Let us denote by $\mathcal{C}$ the set of the polynomial maps $F=H(u_{0},u_{1},\dots,u_{n}):\mathbb{K}^{2}\to\mathbb{K}^{2}$ with $u_{0},u_{1},\dots,u_{n}$ defined in (2.11); and by $\mathcal{C}^{n}_{r,s}$ we denote the corresponding subset of $\mathcal{C}$ for each fixed triple of natural numbers $(r,s,n)$. Theorems 2.18 and 2.20 say that the subsets $\mathcal{C}^{n}_{1,s}$ ($n\geq 1$, $s\geq 1$) and $\mathcal{C}^{1}_{r,s}$ ($r\geq 1$, $s\geq 1$) are the ones satisfying the Jacobian conjecture. Besides, from Proposition 2.23, we have:

Corollary 2.25**.**

If a polynomial map belonging to a class $\mathcal{C}^{n}_{r,s}$ ($n\geq 2,r\geq 2,s\geq 1$)* satisfies the Jacobian condition then it is a counterexample to the Jacobian conjecture.*

3. Two-dimensional complex Jacobian conjecture of degree until 104

We consider now the complex case, i.e. consider polynomial maps $F=(F_{1},F_{2}):\mathbb{C}^{2}\to\mathbb{C}^{2}$. In this section, we will prove that the complex Jacobian conjecture is true for the degree until 104 (Theorem 3.6). For the case of degree 105, we reduce the problem into only four cases to consider (Proposition 3.7).

Let us recall the principal results concerning the complex Jacobian conjecture under the view point of Newton polygons studied by Abhyankar [2], Nagata [17, 18], Appelgate and Onishi [3], Magnus [13], Nakai and Baba [16] and Moh [14]. One can also find the results cited below in the book of van den Essen [8, pages 254-256].

Theorem 3.1** (Abhyankar [2], 1977).**

*If ${\rm deg}\,(F_{1})$ divide ${\rm deg}\,(F_{2})$ or ${\rm deg}\,(F_{2})$ divide

${\rm deg}\,(F_{1})$, then the conjecture is true.*

Let $d:={\rm gcd}({\rm deg}\,(F_{1}),{\rm deg}\,(F_{2}))$ the greatest common divisor of degrees of $F_{1}$ and $F_{2}$.

Theorem 3.2** (Appelgate and Onishi [3], Nagata [17, 18], 1989).**

If $d\leq 8$ or $d$ is a prime number then the conjecture is true.

Theorem 3.3** (Appelgate, Onishi [3], Nagata [18], 1989).**

If ${\rm deg}\,(F_{1})$ or ${\rm deg}\,(F_{2})$ is a product of at most two prime numbers, then the conjecture is true.

Recall that ${\rm deg}\,F={\max}\,\{{\rm deg}\,(F_{1}),{\rm deg}\,(F_{2})\}$. With a highly non-trivial proof, Moh [14] proved:

Theorem 3.4** (Moh [14], 1983).**

The conjecture is true for ${\rm deg}\,(F)\leq 100$.

Using so-called Newton-Puiseux chart, Henryk Żoła̧dek [24] give a stronger result than the one in Theorem 3.2:

Theorem 3.5** (Żoła̧dek [24], 2008).**

The Jacobian conjecture satisfies for maps with ${\rm gcd}({\rm deg}\,(F_{1}),{\rm deg}\,(F_{2}))\leq 16$ and for maps with ${\rm gcd}({\rm deg}\,(F_{1}),{\rm deg}\,(F_{2}))$ equal to 2 times a prime.

In the following, we prove:

Theorem 3.6**.**

The Jacobian conjecture satisfies for ${\rm deg}\,(F)\leq 104$.

Proof.

At first we remark that if ${\rm deg}\,(F_{1})={\rm deg}\,(F_{2})$ then the conjecture is true by Theorem 3.1. Then we can assume that ${\rm deg}\,(F_{1})<{\rm deg}\,(F_{2})$.

1. The case ${\rm deg}\,(F)=101$: since 101 is a prime, the conjecture is true by Theorem 3.2 or by Theorem 3.3.

2. The case ${\rm deg}\,(F)=102$: Assume that ${\rm deg}\,(F_{1})<102$ and ${\rm deg}\,(F_{2})=102$. We see that $102=6.17$ and as $17$ is prime then if ${\rm deg}\,(F_{1})$ is not divisible by 17, the greatest common divisor of ${\rm deg}\,(F_{1})$ and ${\rm deg}\,(F_{2})$ is $d\leq 6$, and the conjecture is true by Theorem 3.2. Then we need to consider only the case ${\rm deg}\,(F_{1})$ is divisible by 17, i.e. the cases ${\rm deg}(F_{1})$ is 17, 34, 51, 68 and 85. If the degree of $F_{1}$ is 17, 34, 51 and 85, then by Theorem 3.3, the conjecture is true since ${\rm deg}\,(F_{1})$ is a product of at most two prime numbers. Now if ${\rm deg}\,(F_{1})=68$, then the greatest common divisor of ${\rm deg}\,(F_{1})$ and ${\rm deg}\,(F_{2})$ is $34$, which is two times a prime, therefore the conjecture is true by Theorem 3.5.

3. The case ${\rm deg}\,(F)=103$: since 103 is a prime, the conjecture is true by Theorem 3.2 or by Theorem 3.3.

4. The case ${\rm deg}\,(F)=104$: Assume that ${\rm deg}\,(F_{1})<104$ and ${\rm deg}\,(F_{2})=104$. Since $104=2^{3}.13$ then if ${\rm deg}\,(F_{1})$ is not divisible by 13, the greatest common divisor of ${\rm deg}\,(F_{1})$ and ${\rm deg}\,(F_{2})$ is $d\leq 8$, and the conjecture is true by Theorem 3.2. Now if ${\rm deg}\,(F_{1})$ is not divisible by 2, then the greatest common divisor of ${\rm deg}\,(F_{1})$ and ${\rm deg}\,(F_{2})$ is $d\leq 13$, the conjecture is also true by Theorem 3.5. Therefore, we need to consider only the case ${\rm deg}\,(F_{1})$ is divisible by 26, i.e. the cases ${\rm deg}(F_{1})$ is $26$, $52$ and $84$. For the two first cases, the conjecture is true by Theorem 3.1. Now if ${\rm deg}\,(F_{1})=84$, the greatest common divisor of ${\rm deg}\,(F_{1})$ and ${\rm deg}\,(F_{2})$ is $26$, which is two times a prime, then the conjecture is also true by Theorem 3.5. ∎

Proposition 3.7**.**

If the conjecture satisfies for the cases where $({\rm deg}\,(F_{1}),{\rm deg}\,(F_{2}))$ is (42, 105), (63, 105), (70, 105) and (84, 105) then the conjecture satisfies for polynomial maps of degree 105.

Proof.

Assume that ${\rm deg}\,(F_{1})<105$ and ${\rm deg}\,(F_{2})=105$. We see that $105=3.5.7$ and as $7$ is prime then if ${\rm deg}\,(F_{1})$ is not divisible by 7, the greatest common divisor of ${\rm deg}\,(F_{1})$ and ${\rm deg}\,(F_{2})$ is $d\leq 15$, and the conjecture is true by Theorem 3.5. Then we need to consider only the case ${\rm deg}\,(F_{1})$ is divisible by 7. We remark that by Theorem 3.1 and Theorem 3.3, the cases we need to consider for ${\rm deg}\,(F_{1})$ are only: 7.4, 7.6, 7.8, 7.9, 7.10, 7.12, 7.14. The cases 7.4, 7.8 and 7.14 satisfy the conjecture by Theorem 3.2 since the greatest common divisor of ${\rm deg}\,(F_{1})$ and ${\rm deg}\,(F_{2})$ in these cases is a prime. So the remain cases are those for which the degree of $F_{1}$ is 42, 63, 70 and 84. ∎

Remark 3.8**.**

Theorems 2.18, Corollary 2.19 and Theorem 2.20 show that the complex conjecture is true for the constructed classes $\mathcal{C}^{n}_{1,s}$ ($n\geq 1$, $s\geq 1$) and $\mathcal{C}^{1}_{r,s}$ ($r\geq 2$, $s\geq 1$) (see also Remark 2.24). Then for studying the conjecture in our constructed class $\mathcal{C}$, we need to study only the maps belonging to the classes $\mathcal{C}^{n}_{r,s}$ ($n\geq 2,r\geq 2,s\geq 1$) (Corollary 2.25). Using non-proper variables of these classes, we can reduce the degree of the map. For example, if $n=2$, $r=2$ and $s=1$, then the non-proper variables in (2.11) are:

[TABLE]

In this case ${\rm deg}(u_{0})=1$, ${\rm deg}(u_{1})=3$ and ${\rm deg}(u_{3})=6$. Then if ${\rm deg}(F)=105$, it is possible to reduce ${\rm deg}(F)$ to only 13, choosing for example $u_{2}u_{3}^{12}$ as a term of the non-proper map $F$. More precisely, each polynomial $F_{1}$ and $F_{2}$ is a finite sum of monomials $u_{0}^{\alpha_{0}}u_{1}^{\alpha_{1}}u_{2}^{\alpha_{2}}$ such that $\alpha_{0}+3\alpha_{1}+6\alpha_{2}\leq 105$. Besides, with a fixed number $n$ of variables, then by increasing the natural numbers $r$ and $s$, we can reduce quickly the degree of the map. **

Statements and Declarations

The author declares that she has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The author was supported by the Research Support Foundation of Sao Paulo State (FAPESP), grants 2023/07802-7 and 2019/21181-0.