A Proof of the Plane Jacobian Conjecture

On the two-dimensional Jacobian conjecture: Magnus' formula revisited, II

This article is part of an ongoing investigation of the two-dimensional Jacobian conjecture. In the first paper of this series, we proved the generalized Magnus' formula. In this paper, inspired by cluster algebras, we introduce a sequence of new conjectures including the remainder vanishing conjecture. This makes the generalized Magnus' formula become a useful tool to show the two-dimensional Jacobian conjecture. In the forthcoming paper(s), we plan to prove the remainder vanishing conjecture.

Plane Jacobian Conjecture for rational polynomials

A non-zero constant Jacobian polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ is invertible if $P$ and $Q$ are rational polynomials.

雅可比猜想的 Tate 代数版本

A Tate algebra version of the Jacobian Conjecture

本文研究配备I-adic拓扑交换环的Tate-雅可比猜想，证明其在特定条件下等价于雅可比猜想、在正特征时不成立，并建立复数域雅可比猜想与C_p上多项式映射逆的Tate代数性质的等价关系。

Jacobian pairs

We study meromorphic jacobian pairs, i.e., pairs of polynomials in one variable, with coefficients meromorphic series in a second variable, whose jacobian relative to the two variables depends only on the second variable. We pose two meromorphic jacobian conjectures about such pairs, one of which is in terms of an invariant of the pair which we call the beta invariant. These conjectures are shown to imply the bivariate algebraic jacobian conjecture which predicts that two bivariate polynomials generate the polynomial ring if their jacobian is a nonzero constant. As another technique for studyi…

Lectures on Geometry and Topology of Polynomials - Surrounding the Jacobian Conjecture

Recent developments of affine algebraic geometry, especially the theory of open algebraic surfaces, provide means to systematically explore geometric and topological properties of polynomials in two variables. Nevertheless, there is one unsurmountable problem remained even in the case of two variables, which has been unsolved since 1939, that is the Jacobian Conjecture. These are notes for author's lectures on the geometry and topology of polynomials and the Jacobian Conjecture delivered at the l'Université de Bordeaux I and at Osaka University in 2003.

关于7次实雅可比猜想的一则注记

A note on the real Jacobian conjecture in degree 7

该研究针对7次实雅可比猜想展开，证明了最高次齐次部分为特定形式的7次多项式对应的实雅可比猜想成立，同时排除了一类非典型雅可比对的存在性。

A note on the plane Jacobian conjecture

It is shown that every polynomial function $P : \mathbb{C}^2\longrightarrow \mathbb{C}$ with irreducible fibres of same a genus is a coordinate. In consequence, there does not exist counterexamples F = (P,Q) to the Jacobian conjecture such that all fibres of P are irreducible curves of same a genus.

一个与 Jacobian 猜想相关的多项式方程组

A system of polynomial equations related to the Jacobian Conjecture

本文将 Jacobian 猜想的真伪归结为某多项式方程组是否有解，并证明其解集为零维。

Some algorithms related to the Jacobian Conjecture

We describe an algorithm that computes possible corners of hypothetical counterexamples to the Jacobian Conjecture up to a given bound. Using this algorithm we compute the possible families corresponding to $\gcd(deg(P),deg(Q))\le 35$, and all the pairs $(deg(P),deg(Q))$ with $\max(deg(P),deg(Q))\le 150$ for any hypothetical counterexample.

Increasing the degree of a possible counterexample to the Jacobian Conjecture from 100 to 108

We list all the pairs $(deg(P),deg(Q))$ with $\max\{deg(P),deg(Q)\}< 125$ for any hypothetical counterexample to the plane Jacobian Conjecture and discard them all, except the pair $(72,108)$ (and the symmetric pair $(108,72)$), thus we confirm the lower bound of 100 obtained by Moh and raise it up to 108.

The Last Aproach to the Settlement of the Jacobian Conjecture

Let S be a polynomial ring over a field of characteristic zero in finitely may variables. Let T be an unramified, finitely generated extension of S with $T^\times = k^\times$. Then T = S.

Comment on a Paper by Yucai Su On Jacobian Conjecture

The said paper [2] entitled "Proof Of Two Dimensional Jacobian Conjecture" is with gaps.

Jacobian Conjecture in two dimension

Let $(P, Q)$ be a pair of Jacobian polynomials. We can show that $ <P, Q>+l+2g(P)-2= 0= <P, [P,Q]>$, where $<f, g>$ is the intersection number of $f, g\in \CC[x, y]$ in the affine plane, $l$ is the number of branch at point at infinity and $g(P)$ is the geometric genus of affine curve defined by $P$. Hence we can show that every Jacobian polynomial defines a smooth rational curve with one point at infinity. It is sufficient to fix the Jacobian conjecture in two dimension by the Abhyankar theorem or the Abhyankar-Moh-Suzuki theorem.

Comment on a Paper by Yucai Su On Jacobian Conjecture (Dec 30, 2005

The said paper [Su2] entitled "Proof Of Two Dimensional Jacobian Conjecture" is false.

Proof of Two Dimensional Jacobian Conjecture

This paper is cancelled

On the two-dimensional Jacobian conjecture: Magnus' formula revisited, I

Let $K$ be an algebraically closed field of characteristic 0. When the Jacobian $({\partial f}/{\partial x})({\partial g}/{\partial y}) - ({\partial g}/{\partial x})({\partial f}/{\partial y})$ is a constant for $f,g\in K[x,y]$, Magnus' formula from [A. Magnus, Volume preserving transformations in several complex variables, Proc. Amer. Math. Soc. 5 (1954), 256--266] describes the relations between the homogeneous degree pieces $f_i$'s and $g_i$'s. We show a more general version of Magnus' formula and prove a special case of the two-dimensional Jacobian conjecture as its application.

关于超平面排列的雅可比理想的两个近期猜想的解决

Closing two recent conjectures related to the Jacobian ideal of hyperplane arrangements

本文解决了Burity-Simis-Tohăneanu关于中心超平面排列的雅可比理想的两个猜想：一个得到证实，另一个在秩至少为4时被反例否定，并揭示了Pfaffian障碍与排列的电路和余维二平坦的关系。

Pencil of irreducible rational curves and Plane Jacobian conjecture

We are concerned with the behavior of the polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ with finite fibres and satisfying the condition that all of the curves $aP+bQ=0$, $(a:b)\in \mathbb{P}^1$, are irreducible rational curves. The obtained result shows that such polynomial maps $F$ is invertible if $(0,0)$ is a regular value of $F$ or if the Jacobian condition holds.

Plane Jacobian conjecture for simple polynomials

A non-zero constant Jacobian polynomial map $F=(P,Q):\mathbb{C}^2 \longrightarrow \mathbb{C}^2$ has a polynomial inverse if the component $P$ is a simple polynomial, i.e. if, when $P$ extended to a morphism $p:X\longrightarrow \mathbb{P}^1$ of a compactification $X$ of $\mathbb{C}^2$, the restriction of $p$ to each irreducible component $C$ of the compactification divisor $D = X-\mathbb{C}^2$ is either degree 0 or 1.

二维Jacobian猜想的一些算术方面

some arithmetical aspects of the two dimensional jacobian conjecture

本文证明了二维Jacobian猜想整数系数反例的渐近值路径仅含有限个整数格点，并应用Theta函数理论于实数域和复数域上的该猜想。

The Jacobian Conjecture as a problem in combinatorics

The Jacobian Conjecture has been reduced to the symmetric homogeneous case. In this paper we give an inversion formula for the symmetric case and relate it to a combinatoric structure called the Grossman-Larson Algebra. We use these tools to prove the symmetric Jacobian Conjecture for the case $F=X-H$ with $H$ homogeneous and $JH^{3}=0$. Other special results are also derived. We pose a combinatorial statement which would give a complete proof the Jacobian Conjecture.

A new method in the Jacobian Conjecture

We introduce a new method in the attempt to prove the Jacobian conjecture. In the complex dimension 2 case, we apply this method to prove some new results related the Jacobian conjecture.

The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture

The Jacobian conjecture in dimension $n$ asserts that any polynomial endomorphism of $n$-dimensional affine space over a field of zero characteristic, with the Jacobian equal 1, is invertible. The Dixmier conjecture in rank $n$ asserts that any endomorphism of the $n$-th Weyl algebra (the algebra of polynomial differential operators in $n$ variables) is invertible. We prove that the Jacobian conjecture in dimension $2n$ implies the Dixmier conjecture in rank $n$. Together with a well-known implication in the opposite direction, it shows that the stable Jacobian and Dixmier conjectures are equi…

雅可比猜想研究中的一个组织原则

An organizing principle in the study of the Jacobian Conjecture

本文通过分析多项式映射的不可约分支，证明了雅可比猜想等价于要么所有映射都是自同构，要么一般映射不是自同构。

Some Comments around The Examples against The Generalized Jacobian Conjecture

We have studied a faded problem, the Jacobian Conjecture ~: \noindent {\sf The Jacobian Conjecture $(JC_n)$}~: If $f_1, \cdots, f_n$ are elements in a polynomial ring $k[X_1, \cdots, X_n]$ over a field $k$ of characteristic $0$ such that the Jacobian $\det(\partial f_i/ \partial X_j) $ is a nonzero constant, then $k[f_1, \cdots, f_n] = k[X_1, \cdots, X_n]$. For this purpose, we generalize it to the following form~: \noindent {\sf The Generalized Jacobian Conjecture $(GJC)$}~: {\it Let $φ: S \rightarrow T$ be an unramified homomorphism of Noetherian domains with $T^\times = φ(S^\times)$. Assume…

A particular case of the Jacobian conjecture

A particular case of the Jacobian conjecture is considered and for small dimensional cases a computational approach is offered

On isomorphisms of factorial domains and the Jacobian conjecture in any characteristic

The main theorem (2.2) consists in two characterizations of isomorphisms of factorial domains in terms of prime or primary rings elements, and unramified, flat or weakly injective affine schemes morphisms. In order to apply this theorem to the famous Jacobian Conjecture, we first introduce its different versions in any characteristic (3.1), and give two reformulations of some these versions in terms of domains of positive characteristic (3.8) and finite prime fields (3.9). Finally, we deduce from the main theorem an original reformulation of the any characteristic version of the Jacobian Conje…

特征 2 下可分雅可比猜想的一个显式反例

An Explicit Characteristic-$2$ Counterexample to the Separable Jacobian Conjecture

在特征为 2 的域上，给出显式多项式自同态 \(F:\mathbb{A}_k^3\to\mathbb{A}_k^3\)，其雅可比行列式为 1 但非单射，次数为 3 构成对可分雅可比猜想反例，稳定化可得各 \(n\geq3\) 维类似反例。

Test polynomials, retracts, and the Jacobian conjecture

Let K[x,y] be the algebra of two-variable polynomials over a field K. A polynomial p=p(x, y) is called a test polynomial (for automorphisms) if, whenever ϕ(p)=p for a mapping ϕof K[x,y], this ϕmust be an automorphism. Here we show that p \in C[x,y] is a test polynomial if and only if p does not belong to any proper retract of C[x,y]. This has the following corollary that may have application to the Jacobian conjecture: if a mapping ϕof C[x,y] with invertible Jacobian matrix is ``invertible on one particular polynomial", then it is an automorphism. More formally: if there is a non-constant poly…

The 0-th Fitting ideal of the Jacobian module of a plane curve

We describe the 0-th Fitting ideal of the Jacobian module of a plane curve in terms of determinants involving the Jacobian syzygies of this curve. This leads to new characterizations of maximal Tjurina curves, that is of non free plane curves, whose global Tjurina number equals an upper bound given by A. du Plessis and C.T.C. Wall.

Fractals and the two dimensional Jacobian Conjecture

We outline an approach to prove the two dimensional Jacobian Conjecture using the theory of fractals.

The ${\rm Jacobian Conjecture}_{2n}$ implies the ${\rm Dixmier Problem}_n$

Using the author's inversion formula for automorphisms of the Weyl algebras with polynomial coefficients and the bound on its degree a slightly shorter (algebraic) proof is given of the result of A. Belov-Kanel and M. Kontsevich that the Jacobian Conjecture implies the Dixmier Proplem. No originality is claimed.

Verification of the Jacobian Conjecture for $d$-linear maps in two variables

For any integer $d \geq 1$, we verify the Jacobian Conjecture for a $d$-linear map in two variables. We prove that almost all the coefficients of the formal inverse are in the ideal specified by the Jacobian condition. We find expressions for certain elements in terms of the generators of this ideal. To obtain these expressions, we generalize a bijective proof of the Cayley-Hamilton theorem in two ways.

满足二维Jacobian猜想的一些类以及复猜想直到次数104的证明

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

本文构造非正常二元多项式映射集，研究Jacobian条件，获得满足二维猜想的类，并用Newton多边形技术将复猜想证明推进到次数104，超越Moh边界。

Holonomic modules and 1-generation in the Jacobian Conjecture

A polynomial endomorphism $σ\in {\rm End}_K(P_n)$ is called a Jacobian map if its Jacobian is a nonzero scalar (the field has zero characteristic). Each Jacobian map $σ$ is extended to an endomorphism $σ$ of the Weyl algebra $A_n$. The Jacobian Conjecture (JC) says that every Jacobian map is an automorphism. Clearly, the Jacobian Conjecture is true iff the twisted (by $σ$) $P_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. It is shown that the $A_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. Furthermore, the $A_n$-module ${}^σ P_n$ is holonomic and as a result h…

Jacobian Conjecture and Nilpotent Mappings

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).

A proof of the Jacobian conjecture

Based on the results people have obtained, we try to prove the Jacobian conjecture, but there is a gap in the proof.

Plane Jacobian problem for rational polynomials

This paper has been withdrawn by the author due to a crucial error in the last lines in the proof of Lemma 3.3.

Some Remarks on the Jacobian Conjecture and Connections with Hilbert's Irreducibility Theorem

We make two observations regarding the invertibility of Keller maps. i.e., polynomial maps for which the determinant of their Jacobian matrix is identically equal to 1. In our first result, we show that if P is a n-dimensional Keller map, defined over any extension of Q, then P has a polynomial inverse if and only if the range of P contains the cartesian product of n universal Hilbert sets. In our second result, we show that if P is a 2-dimensional Keller map, defined over any algebraic number field, then P is invertible on a set that contains almost all rational integers of K.

The 2-dimensional jacobian conjecture via Klein's program

We investigate the 2-dimensional jacobian conjecture via Klein's program.

Keller映射局部双射性的推广及二维Jacobian猜想的证明

Generalizations of local bijectivity of Keller maps and a proof of $2$-dimensional Jacobian conjecture

本文通过推广并运用Keller映射的局部双射性，证明了二维Jacobian猜想。

寻找雅可比猜想反例的一种初等方法

An elementary way to find a counterexample to the Jacobian Conjecture

本文提出一种初等数学方法，用于构造雅可比猜想的反例，所得反例与Levent Alpöge的已知例子在坐标变换下等价。

Jacobian conjecture as a problem on integral points on affine curves

It is shown that the $n$-dimensional Jacobian conjecture over algebraic number fields may be considered as an existence problem of integral points on affine curves. More specially, if the Jacobian conjecture over $\mathbb{C}$ is false, then for some $n\gg 1$ there exists a counterexample $F\in \mathbb{Z}[X]^n$ of the form $F_i(X)=X_i+ (a_{i1}X_1+\dots+a_{in}X_n)^{d_i}$, $a_{ij}\in \Z$, $d_i=2;3 $, $i,j=\overline{1,n},$ such that the affine curve $F_1(X)=F_2(X)=\dots=F_n(X)$ has no non-zero integer points.

A geometric approach to the two-dimensional Jacobian Conjecture

Any counterexample to the two-dimensional Jacobian Conjecture gives a rational map from one projective plane to another. We use some ideas of the Minimal Model Program to study the combinatorial structure of a rational surface, that is obtained by resolving this map. Several structural results are proven, revealing a rather orderly structure of the graph of the curves at infinity. We also exhibit and discuss a graph that may lead to a counterexample to the Jacobian Conjecture.

The Jacobian Conjecture: Approximate roots and intersection numbers

We translate the results of Yansong Xu into the language of~\cite{GGV1}, obtaining nearly the same formulas for the intersection number of Jacobian pairs, but with an inequality instead of an equality.

论二维 Jacobian 猜想：重访 Magnus 公式，IV

On the two-dimensional Jacobian conjecture: Magnus' formula revisited, IV

本文围绕二维 Jacobian 猜想重审广义 Magnus 公式，通过研究 $F$、$G$ 的内部多项式，证明其 Newton 多边形东北顶点位于特定区域，并据此提出及在特例中证明相关猜想。

On the Jacobian Conjecture and ideal membership for degree $d$-linear maps

We consider polynomial maps, which we call degree $d$-linear maps, that satisfy the Jacobian condition. We prove that certain infinite families of elements, which appear in the coefficients of the formal inverse of such maps, are in the ideal determined by the Jacobian condition. Using the Cayley-Hamilton theorem, we provide expressions for these elements in terms of the generators of that ideal. We also give a combinatorial proof of the Cayley-Hamilton theorem similar to that of Straubing \cite{Straubing}. We also include results of Gröbner basis computations regarding other elements.

On the Rational Real Jacobian Conjecture

Jacobian conjectures (that nonsingular implies a global inverse) for rational everywhere defined maps of real n-space to itself are considered, with no requirement for a constant Jacobian determinant or a rational inverse. The birational case is proved and the Galois case clarified. Two known special cases of the Strong Real Jacobian Conjecture (SRJC) are generalized to the rational map context. For an invertible map, the associated extension of rational function fields must be of odd degree and must have no nontrivial automorphisms. That disqualifies the Pinchuk counter examples to the SRJC a…

Jacobian代数的分层Betti数与平面曲线的总Tjurina数

Graded Betti numbers of the Jacobian algebra and total Tjurina numbers of plane curves

本文通过Jacobian代数的分层Betti数计算平面曲线的总Tjurina数，并提出新的上界视角及必要条件。

Infinite-dimensional algebraic varieties and proof of the Jacobian Conjecture

This paper has been withdrawn by the author, due an error in the proof of Proposion 2.13.

A proof of the Generalized Jacobian conjecture

Based on the reduction of degree in polynomial mappings and some known results in algebraic geometry, by introducing the Brouwer degree, a tool from differential topology, algebraic topology and algebraic geometry, we completely prove the Generalized Jacobian conjecture in the field of real numbers, which implies the Generalized complex Jacobian conjecture. Also, for the strong real Jacobian conjecture, we present a newly sufficient and necessary condition.

Finite determination conjecture for Mather-Jacobian minimal log discrepancies and its applications

In this paper we study singularities in arbitrary characteristic. We propose Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in terms of jet schemes of a singularity. The conjecture is equivalent to the boundedness of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy. We also show that this conjecture yields some basic properties of singularities; e.g. openness of Mather-Jacobian (log) canonical singularities, stability of these singularities under small deformations and lower semi-continuity of Mather-…

Jacobian conjecture in $\mathbb R^2$

Jacobian conjecture states that if $F:\ \mathbb C^n(\mathbb R^n)\rightarrow \mathbb C^n(\mathbb R^n)$ is a polynomial map such that the Jacobian of $F$ is a nonzero constant, then $F$ is injective. This conjecture is still open for all $n\ge 2$, and for both $\mathbb C^n$ and $\mathbb R^n$. Here we provide a positive answer to the Jacobian conjecture in $\mathbb R^2$ via the tools from the theory of dynamical systems.

Saturation of Jacobian ideals: some applications to nearly free curves, line arrangements and rational cuspidal plane curves

In this note we describe the minimal resolution of the ideal $I_f$, the saturation of the Jacobian ideal of a nearly free plane curve $C:f=0$. In particular, it follows that this ideal $I_f$ can be generated by at most 4 polynomials. Related general results by Hassanzadeh and Simis on the saturation of codimension 2 ideals are discussed in detail. Some applications to rational cuspidal plane curves and to line arrangements are also given.