A note on Kaliman's weak Jacobian Conjecture

1.   Introduction

Let $\mathbb{C}[X_1, \cdots, X_n]$ denote the polynomial ring in the variables $X_1, \cdots, X_n$ over $\mathbb{C}.$ A polynomial map is a map $F = (F_1, \cdots, F_n): \mathbb{C}^n \to \mathbb{C}^n$ of the form

where each $F_i$ belongs to $\mathbb{C}[X_1, \cdots, X_n].$ Such a polynomial map is called invertible if there exists a polynomial map $G = (G_1, \cdots, G_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ such that $X_i = G_i(F_1, \cdots, F_n)$ for all $1\leq i\leq n,$ i.e., $G$ is the left inverse of $F.$ It is easy to show that $G$ is also a right inverse of $F.$ So $F$ is invertible, i.e., $F$ is an isomorphism, in the sense of morphisms of algebraic varieties.

Consider a polynomial map $F: \mathbb{C}^n\rightarrow \mathbb{C}^n$. How can we recognize if a polynomial map $F$ is invertible?

Let $J(F) = (\partial F_i/\partial X_j)$ be the Jacobian matrix of $F$. Clearly, the invertibility of the matrix $J(F)$ is equivalent to det$J(F)\in \mathbb{C}^\times.$ It is easy to show that if $F: \mathbb{C}^n\rightarrow \mathbb{C}^n$ is invertible, then det$J(F)\in \mathbb{C}^\times.$ Conversely, there is the following famous conjecture.

Conjecture 1.1. If $\textrm{det}$$J(F) \in \mathbb{C}^\times,$ then $F$ is invertible.

The Jacobian Conjecture was first formulated by O. H. Keller in 1939. Aside from the trivial case $n = 1,$ this conjecture remains an open problem for all $n\geq2$ up to now. The Jacobian Conjecture appeared as Problem 16 on a list of 18 famous open problems in the paper by Steve Smale ^[11]. The Jacobian Conjecture has been reduced to the case of degree 3 using the method of algebraic $K$-theory by Bass, Connell, and Wright ^[1]. The second author has achieved some results in algebraic $K$-theory ^[12,13].

When $n = 2$, Kaliman proposed the weak Jacobian conjecture in ^[6].

Conjecture 1.2. Let $F = (F_1, F_2): \ \mathbb{C}^2\to \mathbb{C}^2$ with $\textrm{det}$$J(F)\in \mathbb{C}^\times.$ Suppose that for every $c\in \mathbb{C}$ the fibre $V(F_{1}): = \{(x, y)\mid F_1(x, y) = c\}$ is irreducible. Then the map $F$ is invertible.

The fiber $V(F_{1})$ is irreducible if and only if the polynomial $F_1(x, y)-c$ is irreducible. For a polynomial $f(x, y)\in \mathbb{C}[x, y]$ with the degree deg$f(x, y) > 1$, in general the polynomial $f(x, y)-c$ is not always irreducible for each $c\in\mathbb{C}$. Hence, our main improvement is the following theorem (see Section 2, Thm.2.8)

Theorem 1.3. Let $F = (F_1, F_2): \ \mathbb{C}^2\to \mathbb{C}^2$ with $\textrm{det}$$J(F)\in \mathbb{C}^\times.$ Suppose that there exist infinitely many $c\in \mathbb{C}$ such that the polynomial $F_1(x, y)-c$ is irreducible. Then the map $F$ is invertible.

Furthermore, we give a general form of the above theorem

Theorem 1.4. Let $F = (F_1, F_2): \ \mathbb{C}^2\to \mathbb{C}^2$ with $\textrm{det}$$J(F)\in \mathbb{C}^\times.$ If there exist infinitely many points $(a, b, c)\in \mathbb{C}^3$ such that the polynomial $aF_1(x, y)+bF_2(x, y)+c$ is irreducible, then the map $F$ is invertible.

In the above theorem, the condition that $aF_1(x, y)+bF_2(x, y)+c$ is irreducible can be independent of the Jacobian conjecture. This leads us to propose the following conjecture.

Conjecture 1.5. Let $F_1(x, y), F_2(x, y)\in \mathbb{C}[x, y]$ be algebraically independent polynomials. Then there exist infinitely many points $(a, b, c)\in \mathbb{C}^3$ such that the polynomial $aF_1(x, y)+bF_2(x, y)+c$ is irreducible.

There are many works on the case $n = 2$. A good introduction about the classical results can be found in chapter 10 in ^[4]. Miyanishi ^[8] proved that the Jacobian conjecture holds true if a generalized Sard property holds true for the affine plane and an $\mathbb{A}^1$-fibration on $\mathbb{A}^2$. Jedrzejewicz and Zieliński in ^[5] give a survey of a new purely algebraic approach to the Jacobian Conjecture in terms of irreducible elements and square-free elements. A similar result has been achieved in ^[2,3]. However, our methods are based on the Hurwitz formula and resolution of the singular curve.

2.   Proof of theorem

Let $F = (F_1, F_2): \ \mathbb{C}^2\to \mathbb{C}^2$ be a polynomial map such that det$J(F) \in \mathbb{C}^\times$. Denote $m = \max\{\text{deg}F_1, \text{deg}F_2\}$ the maximal degree of $F_1$ and $F_2$. Then we have a rational map of projective spaces

where $\bar{F}_i(x, y, z) = z^mF_i(\frac{x}{z}, \frac{y}{z})$ are the homogeneous polynomials. Let

Then $\mathbb{P}^2_\mathbb{C} = \mathbb{C}^2\bigcup \mathbb{P}^1_\mathbb{C}$. Moreover, the restriction of $\bar{F}$ on $\mathbb{C}^2$ is $F$ and

Lemma 2.1. Let $F = (F_1, F_2):\ \mathbb{C}^2\to \mathbb{C}^2$ be a polynomial map such that $\textrm{det}$$J(F)\neq 0$ in $\mathbb{C}[x, y]$. Then $F_1, F_2$ are algebraically independent over $\mathbb{C}$ and $\mathbb{C}(F_1, F_2)\subset\mathbb{C}(x, y)$ is a finite field extension.

Proof. A proof can be found in [4, Prop.1.1.31]. □

Lemma 2.2. For the polynomial map $F = (F_1, F_2): \ \mathbb{C}^2\to \mathbb{C}^2$, if $\textrm{det}$$J(F)\neq 0$ in $\mathbb{C}[x, y]$, then the cardinality of the fibers of $F$ is bounded by the degree

Proof. See [4, Thm.1.1.32]. □

Lemma 2.3. Let $F = (F_1, F_2): \ \mathbb{C}^2\to \mathbb{C}^2$ be the polynomial map such that $\textrm{det}$$J(F)\neq 0$ in $\mathbb{C}[x, y]$. Then there exists a Zariski open set $U\subset \mathbb{C}^2$ such that

Proof. See [9, Prop.3.17]. The condition det$J(F)\neq 0$ ensures that the map $F$ is dominating map. □

Lemma 2.4. Let $F = (F_1, F_2): \ \mathbb{C}^2\to \mathbb{C}^2$ be the polynomial map such that $\textrm{det}$$J(F)\in \mathbb{C}^\times$. Denote $V(F_1) = \{(x, y)\in \mathbb{C}^2\mid F_1(x, y) = c, \ c\in\mathbb{C}\}$ and $L_c = \{(x, y)\in \mathbb{C}^2\mid x = c\}$. Then the morphism

is étale morphism.

Proof. First, the morphism $F$ is étale. Let $L_c = \{(x, y)\in \mathbb{C}^2\mid x = c\}$. Then $V(F_1) = F^{-1}(L_c)$. Hence we have the fiber product

Because the étale map is stable under fibered products (see [7, Chap.4, Prop.3.22]), the map $F_2:\ V(F_1)\to L_c$ is étale. □

Lemma 2.5. Let $F = (F_1, F_2): \ \mathbb{C}^2\to \mathbb{C}^2$ be the polynomial map such that $\textrm{det}$$J(F)\neq 0$. Denote $V(F_1)$ and $L_c$ as in Lemma 2.4. Consider

Then there exists a Zariski open set $U\subset\mathbb{C}^2$ such that for almost all $c\in\mathbb{C}$,

Proof. By Lemma 2.3, $\mathbb{C}^2\setminus U$ contains at most finitely many lines $L_c$. Hence the lemma follows: □

Lemma 2.6. Let $F = (F_1, F_2): \ \mathbb{C}^2\to \mathbb{C}^2$ be the polynomial map such that $\textrm{det}$$J(F)\in \mathbb{C}^\times$. Denote $m = \textrm{deg}F_1$ and $\bar{F}_1(x, y, z) = z^mF_1(\frac{x}{z}, \frac{y}{z})$. Let $V(\bar{F}_1): = \{(x:y:z)\in\mathbb{P}^2_\mathbb{C}\mid \bar{F}_1(x, y, z) = 0\}$ and $L_\infty = \{(x:y:z)\in\mathbb{P}^2_\mathbb{C}\mid z = 0\}$. Then the curve $V(\bar{F}_1)$ is smooth at $V(\bar{F}_1)\setminus F^{-1}(L_\infty)$. The set $V(\bar{F}_1)\bigcap F^{-1}(L_\infty)$ may be singular points of $V(\bar{F}_1)$.

Proof. This lemma is easy, because $V(\bar{F}_1)\setminus F^{-1}(L_\infty) = V(F_1)$ and $V(F_1)$ is smooth. □

Lemma 2.7. $\textrm{(Hurwitz)}$ Let $\phi: C_1\to C_2$ be a morphism of Riemann surfaces of genera $g_1$ and $g_2$. Then

where $\textrm{deg}$$\phi$ is the degree of the map $\phi$, $e_{\phi}(P)$ is the ramification index of $\phi$ at $P$.

Proof. A proof can be found in [10, Thm.5.9]. □

Let $m = \text{deg}F_1(x, y)$. Then $\bar{F}_1(x, y, z) = z^mF_1(\frac{x}{z}, \frac{y}{z})$ is an irreducible polynomial if and only if $F_1(x, y)$ is irreducible. Now we can prove the main theorems in this section.

Theorem 2.8. Let $F = (F_1, F_2): \ \mathbb{C}^2\to \mathbb{C}^2$ be the polynomial map such that $\textrm{det}$$J(F)\in \mathbb{C}^\times$. Suppose that there exist infinitely many $c\in \mathbb{C}$ such that the polynomial $F_1(x, y)-c$ is irreducible. Then the two-dimensional Jacobian Conjecture holds.

Proof. Let $\bar{F}_1(x, y, z) = z^mF_1(\frac{x}{z}, \frac{y}{z})-cz^m$. The projective set $V(\bar{F}_1)\subset \mathbb{P}_\mathbb{C}^2$ is defined by $\bar{F}_1(x, y, z) = 0$ and $L_c\subset \mathbb{P}_\mathbb{C}^2$ defined by $x = c$. Consider the map

Since there exist infinitely many $c\in \mathbb{C}$ such that the polynomial $F_1(x, y)-c$ is irreducible, we can find some $c\in\mathbb{C}$ such that $V(\bar{F}_1)$ is irreducible and satisfying Lemma 2.5, that is, deg$\phi_c = \text{deg}F$. Further, $\phi_c$ is étale restricting on the affine curve $V(F_1)$ by Lemma 2.4, where $V(F_1)$ is the affine part of $V(\bar{F}_1)$.

If $V(\bar{F}_1)$ is singular at $V(\bar{F}_1)\setminus V(F_1) = \phi_c^{-1}(\infty)$, where $\infty = (c:1:0)\in L_c$, then from resolution of singularity, we can find a smooth curve $\mathcal{C}$ such that the morphism

satisfying that $r$ is isomorphic on $W: = r^{-1}(V(F_1))$ (see [9, Chp.7, P.128]). Then we have

is étale on $W$.

Since the genus of $L_c$ is 0, by Lemma 2.7,

where $g$ is the genus of $\mathcal{C}$. Since $\phi$ is étale on $W$, we have $e_{\phi}(P) = 1$ for $P\in W$. But $\mathcal{C}\setminus W = \phi^{-1}(\infty)$, by Proposition 2.6 in ^[10], we have

Hence,

Since deg$\phi\geq 1, \ \#\phi^{-1}(\infty)\geq 1$, the right side in the above is negative. Then $2g-2 < 0$, therefore we have $g = 0$. Furthermore,

This implies that $F$ is injective. By Theorem 4.1.1 in ^[4], $F$ is isomorphic. □

Theorem 2.9. Let $F = (F_1, F_2): \ \mathbb{C}^2\to \mathbb{C}^2$ with $\textrm{det}$$J(F)\in \mathbb{C}^\times.$ If there exist infinitely many points $(a, b, c)\in \mathbb{C}^3$ such that the polynomial $aF_1(x, y)+bF_2(x, y)+c$ is irreducible, then the map $F$ is invertible.

Proof. The proof of this theorem is similar to Theorem 2.8, because $ax+by+c = 0$ defines a line in $\mathbb{C}^2$, which is isomorphic to $\mathbb{C}^1$. □

3.   Irreducibility of polynomials

Let $F(x, y)\in\mathbb{C}[x, y]$ such that $F(x, y)\not\in \mathbb{C}[x]$ nor $F(x, y)\not\in \mathbb{C}[y]$. We check Conjecture 1.5 when deg$F = 2$.

Proposition 3.1. Conjecture 1.5 holds true when deg$F = 2$.

Proof. Let $F(x, y) = ax^2+bxy+cy^2+dx+ey+f\in \mathbb{C}[x, y]$. Since deg$F = 2$, at least one of $a, b, c$ is not 0, we can assume $a\neq 0$. Consider $F(x, y)/a$. Then we can assume $a = 1$. Since $F(x, y)\not\in \mathbb{C}[x], F(x, y)\not\in\mathbb{C}[y]$, we discuss it in several cases.

Case 1. $b = c = 0, e\neq0$. Then for each $z\in \mathbb{C}$, $F(x, y)+z$ is irreducible.

Case 2. At least one of $b, c$ is not 0. Supposing for some $z\in \mathbb{C}$ there is

Comparing the homogeneous part of degree 2, we have

Comparing the homogeneous part of degree 1, we have

If $a_1-b_1 = \sqrt{b^2-4c}\neq 0$, then the above equation has a unique solution for $a_2, b_2$. Hence, there exists only one $z\in\mathbb{C}$ such that $F(x, y)+z$ is reducible.

If $a_1-b_1 = \sqrt{b^2-4c} = 0$, then $a_1\neq 0$; otherwise, we have $b = c = 0$, a contradiction. Then the above equation becomes

If $d\neq e/a_1$, then there exists no solution for the above equation. Hence, for each $z\in \mathbb{C}$, $F(x, y)+z$ is irreducible. □

4.   Conclusions

In this paper, we generalize Kaliman's weak Jacobian Conjecture utilizing the Hurwitz formula and resolution of singular curves. At the same time, we give a conjecture about the property of irreducibility of linear combination polynomials in two variables. Furthermore, we check this conjecture in the case of polynomials with degree 2.

Author contributions

Yan Tian: conceptualization, formal analysis, investigation, methodology, writing-original draft, writing-review and editing, funding acquisition; Chaochao Sun: supervision, methodology, formal analysis, writing-review and editing. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

Department of Education University-Industry Collaborative Education Program. Project number: 230804092233033.

Conflict of interest

The authors declare no conflict of interest.