An Approach to the Jacobian Conjecture

On the shape of possible counterexamples to the Jacobian Conjecture

We improve the algebraic methods of Abhyankar for the Jacobian Conjecture in dimension two and describe the shape of possible counterexamples. We give an elementary proof of the result of Heitmann, which states that gcd(deg(P),deg(Q)) is greater than or equal to 16 for any counterexample (P,Q). We also prove that gcd(deg(P),deg(Q)) \ne 2p for any prime p and analyze thoroughly the case 16, adapting a reduction of degree technique introduced by Moh.

The two-dimensional Jacobian conjecture and the lower side of the Newton polygon

We prove that if the Jacobian Conjecture in two variables is false and (P,Q) is a standard minimal pair, then the Newton polygon HH(P) of P must satisfy several restrictions that had not been found previously. This allows us to discard some of the corners found in [GGV, Remark 7.14] for HH(P), together with some of the infinite families found in [H, Theorem~2.25]

Jacobian Conjecture的特征p方法

Characteristic p approaches to the Jacobian Conjecture

本文提出了正特征下Jacobian Conjecture的多个版本，经多族Jacobian pair测试后，提出了一种解决特征0下Jacobian Conjecture的特征p方法。

The two-dimensional Centralizer Conjecture

A result by C. C.-A. Cheng, J. H. Mckay and S. S.-S. Wang says the following: Suppose the Jacobian of $A$ and $B$ is invertible in $\mathbb{C}[x,y]$ and the Jacobian of $A$ and $w$ is zero for $A,B,w \in \mathbb{C}[x,y]$. Then $w \in \mathbb{C}[A]$. We show that in CMW's result it is possible to replace $\mathbb{C}$ by any field of characteristic zero, and we conjecture the following 'two-dimensional Centralizer Conjecture over $D$': Suppose the Jacobian of $A$ and $B$ is invertible in $D[x,y]$ and the Jacobian of $A$ and $w$ is zero for $A,B,w \in D[x,y]$, $D$ is an integral domain of charact…

评Kuo，Parusinski和Paunescu关于Jacobian猜想的文章

Comment on a Paper by Kuo, Parusinski and Paunescu On Jacobian Conjecture

该论文题为“一个证明的平面雅可比猜想”是不正确的。

On Jacobian conjecture

withdrawed due to a substantial error.

关于Jacobian猜想

On The Jacobian Conjecture

这篇文章已被作者撤回。

An approach to the Jacobian Conjecture in terms of irreducibility

We present some motivations and discuss various aspects of an approach to the Jacobian Conjecture in terms of irreducible elements and square-free elements.

与雅可比猜想相关的一个多项式系统的Gröbner基与解集

The Groebner basis and solution set of a polynomial system related to the Jacobian conjecture

本文针对与雅可比猜想相关的多项式系统，计算其Gröbner基并完整刻画解集，为研究该猜想提供了代数计算依据。

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.

评苏育才关于雅可比猜想的一篇文章

Comment on a Paper by Yucai Su On Jacobian Conjecture

上述论文[2]题为“二维雅可比猜想的证明”是有差距的。

评苏育才关于Jacobian猜想的论文（2005年12月30日

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

上述文件[苏2]题为“证明二维雅可比猜想”是假的。

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。

平面Jacobian猜想的一个证明

A Proof of the Plane Jacobian Conjecture

由于第7节中的错误，本文已被作者撤回。

二维Jacobian猜想的证明

Proof of Two Dimensional Jacobian Conjecture

这篇论文被取消了

雅可比对

Jacobian pairs

我们研究亚纯雅可比对，即，一个变量的多项式对，系数亚纯级数在第二个变量，其雅可比相对于两个变量只取决于第二个变量。我们提出了两个亚纯雅可比代数关于这样的对，其中之一是在一个不变的对，我们称之为β不变量。证明了二元代数雅可比猜想的存在，该猜想预言了如果两个二元多项式的雅可比是一个非零常数，则它们生成多项式环。作为研究雅可比猜想的另一种技术，我们重新审视牛顿多边形。

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.

A special case of the two-dimensional Jacobian Conjecture

Let $f: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be a $\mathbb{C}$-algebra endomorphism having an invertible Jacobian. We show that for such $f$, if, in addition, the group of invertible elements of $\mathbb{C}[f(x),f(y),x][1/v] \subset \mathbb{C}(x,y)$ is contained in $\mathbb{C}(f(x),f(y))-0$, then $f$ is an automorphism. Here $v \in \mathbb{C}[f(x),f(y)]-0$ is such that $y = u/v$, with $u \in \mathbb{C}[f(x),f(y),x]-0$. Keller's theorem (in dimension two) follows immediately, since Keller's condition $\mathbb{C}(f(x),f(y))=\mathbb{C}(x,y)$ implies that the group of invertible elements of $\math…

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.

The Gaussian Moments Conjecture and the Jacobian Conjecture

We first propose what we call the Gaussian Moments Conjecture. We then show that the Jacobian Conjecture follows from the Gaussian Moments Conjecture. We also give a counter-example to a more general statement known as the Moments Vanishing Conjecture .

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。

组合学中的雅可比猜想

The Jacobian Conjecture as a problem in combinatorics

雅可比猜想被简化为对称齐次情形。本文给出了对称情形的一个反演公式，并将它与一种称为Grossman-Larson代数的组合结构联系起来。我们使用这些工具来证明对称雅可比猜想的情况下$F=X-H$与$H$齐次和$JH^{3}=0$。还得到了其它一些特殊的结果。我们提出一个组合命题，它将给出一个完整的证明雅可比猜想.

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.

Jacobian猜想与Dixobian猜想是稳定等价的

The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture

关于$n$维的雅可比猜想断言，在特征为零的域上，$n$维仿射空间的任何多项式自同态，当雅可比等于1时，都是可逆的。秩为$n$的Dixandian猜想断言$n$阶Weyl代数（$n$变量中的多项式微分算子代数）的任何自同态都是可逆的。证明了维度$2n$上的Jacobian猜想蕴含秩$n$上的Dixmier猜想。再加上一个著名的相反方向的含义，它表明稳定的雅可比矩阵和Dixobian矩阵是等价的。证明的主要工具是有限特征化约。在论文完成后，我们了解到主要结果已经由Y.Tsuchimoto在Osaka Journal of Mathematics第42卷第2期（2005年6月）上发表。他的证明是不同的。

解决Jacobian猜想的最后途径

The Last Aproach to the Settlement of the Jacobian Conjecture

设S是特征为零的域上的多项式环，其变量为N.设T是S的一个非分歧的、非生成的扩张，其上有$T^\times = k^\times$ .则T = S。

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 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.

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。

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.

${\rm Jacobian Conjecture}_{2n}$意味着${\rm Dixmier Problem}_n$

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

利用作者给出的多项式系数Weyl代数自同构的反演公式及其次数的界，对A. Belov-Kanel和M. Kontsevich证明了Jacobian猜想蕴含着Dixobian问题。没有原创性。

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.

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.

简单多项式的平面Jacobian猜想

Plane Jacobian conjecture for simple polynomials

非零常数雅可比多项式映射$F=(P,Q):\mathbb{C}^2 \longrightarrow \mathbb{C}^2$具有多项式逆，如果分量$P$是简单多项式，即，如果当$P$扩展到$\mathbb{C}^2$的紧致化$X$的态射$p:X\longrightarrow \mathbb{P}^1$时，$p$对紧致化因子$D = X-\mathbb{C}^2$的每个不可约分量$C$的限制是次数0或1。

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…

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.

关于阶乘整环的同构与任意特征的Jacobian猜想

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

主要定理（2.2）是关于阶乘整环的素环或准素环元同构和非分歧、平坦或弱内射仿射概型态射的两个刻画。为了将这一定理应用于著名的雅可比猜想，我们首先介绍了它在任何特征上的不同版本（3.1），并给出了一些这些版本在正特征域（3.8）和有限素域（3.9）上的两个重新表述。最后，我们从主要定理中推出了雅可比猜想的任何特征形式在素环或准素环元素（3.11）方面的原始重新表述。

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。

测试多项式，收缩和雅可比猜想

Test polynomials, retracts, and the Jacobian conjecture

设K [x，y]是域K上二元多项式代数。一个多项式p = p（x，y）称为一个测试多项式（对于自同构），如果，每当对于K [x，y]的一个映射，（p）= p时，这个一定是一个自同构。本文证明了p \in C [x，y]是一个检验多项式当且仅当p不属于C [x，y]的任何真收缩核.这有以下推论，可以应用于雅可比猜想：如果C [x，y]的映射具有可逆的雅可比矩阵"在一个特定的多项式上是可逆的"，那么它是自同构。更正式地说：如果存在一个非常数多项式p和C [x，y]的一个内射映射使得（（p））= p，则是一个自同构。

The inverse function theorem and the resolution of the Jacobian conjecture in free analysis

We establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices. We show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity, the function must be invertible. Thus, as a corollary, we establish the Jacobian conjecture in this context. Furthermore, our result holds for commutative polynomials evaluated on tuples of commuting matrices.

作为微扰量子场论问题的雅可比猜想

The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory

雅可比猜想是一个古老的数学未解问题，人们从不同的角度对它进行了攻击，但都没有成功。我们在这里添加另一个观点有关的所谓正式的逆方法，即微扰量子场论。

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.

关于Jacobian猜想的几点注记及其与Hilbert不可约定理的联系

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

我们对Keller映射的可逆性作了两个观察。也就是说，多项式映射，其雅可比矩阵的行列式同等于1。在我们的第一个结果中，我们证明了如果P是定义在Q的任意扩张上的n维Keller映射，则P有多项式逆当且仅当P的值域包含n个泛Hilbert集的笛卡儿积.在第二个结果中，我们证明了如果P是定义在任何代数数域上的二维Keller映射，则P在包含K的几乎所有有理整数的集合上可逆。

关于仿射空间在仿射簇中的开嵌入及雅可比猜想

On Open Embeddings of Affine Spaces in Affine Varieties and the Jacobian Conjecture $(JC_2)$

本文探讨了仿射空间在仿射簇中的开嵌入问题，并通过拓扑方法证明了雅可比猜想及相关推广猜想，揭示了在特征为0的域上，开嵌入与雅可比猜想之间的联系。

多项式收缩与雅可比猜想

Polynomial Retracts and the Jacobian Conjecture

设$ K[x, y]$是特征为$0$的域$K$上的二元多项式代数。$K[x, y]$的一个子代数$R$称为收缩，如果存在一个幂等同态（{ \it收缩}或{ \it投影}）$φ: K[x, y] \to K[x, y]$使得$φ(K[x, y]) = R$。其他等价的收缩定义的存在提供了几种不同的研究方法，并汇集了组合代数，同调代数和代数几何的思想。在本文中，我们刻画了$ K[x, y]$的所有收缩到一个自同构，并给出了这个刻画的几个应用，特别是著名的雅可比猜想。值得注意的是，我们证明了如果$K[x,y]$的多项式映射$φ$具有可逆雅可比矩阵{ \it和}固定一个非常数多项式，则$φ$是自同构。

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。

The Jacobian Conjecture fails for pseudo-planes

A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its étale endomorphisms are proper. We study the conjecture for $\mathbb{Q}$-acyclic surfaces of negative Kodaira dimension. We show that $G$-equivariant counterexamples for infinite group $G$ exist if and only if $G=\mathbb{C}^*$ and we classify them relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected $\mathbb{C}^*$-surfaces of negative Kodaira dimension which admit non-proper $\mathbb{C}^*$-equivariant étale endomorphisms. We prove also that for every integers $r\geq …

实Jacobian猜想中的一个尖锐次数界

A sharp degree bound in the real Jacobian conjecture

本文证明了当多项式映射F的p的次数为6时F是单射，并构造了次数为7的非单射例子，从而确定了保证全局单射性的最优次数界为6。

关于黑塞猜想的一个五变量反例，以及雅可比猜想和黑塞猜想的低维状态

A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures

本文给出一个五变量整数多项式\(\Psi\)作为黑塞猜想\(\HC_5\)的反例，通过单变量舒尔下降从阿尔波格的六变量雅可比反例翻倍得到。结合相关定理及反驳，确定了除\(n = 4\)外各维度黑塞猜想的真假，还给出六变量\(\HC_6\)反例，扩展并取代了作者早期笔记。