Frameworks for two-dimensional Keller maps

齐次Keller映射的碰撞-外壳压缩与赵消失猜想的四十变量反例

Collision-Generated Compression for Homogeneous Keller Maps

该研究提出齐次Keller映射的碰撞生成压缩原理，利用极化代数构造子代数，基于Thompson的24变量映射得到20维子空间，构造出赵消失猜想的四十变量反例并验证相关计算。

分次凯勒映射与雅可比猜想

Graded Keller maps and the Jacobian Conjecture

探讨加权射影空间有理点稀疏观点对雅可比猜想的意义，证明分次符号模式决定一切，如权重全正等变凯勒映射是自同构，二维任意符号模式也成立，还阐述了凯勒条件在商空间情况及相关轨迹情况，提出按权重签名分类等变多项式映射。

A characterization of Keller maps

Let k be a field of characteristic zero. Let phi be a k-endomorphism of the polynomial algebra k[x_1,...,x_n]. It is known that phi is an automorphism if and only if it maps irreducible polynomials to irreducible polynomials. In this paper we show that phi satisfies the jacobian condition if and only if it maps irreducible polynomials to square-free polynomials. Therefore, the Jacobian Conjecture is equivalent to the following statement: every k-endomorphism of k[x_1,...,x_n], mapping irreducible polynomials to square-free polynomials, maps irreducible polynomials to irreducible polynomials.

The intrinsic topology of linear maps

Based on many experts' former work in the Jacobian conjecture and an essential analysis of intrinsic topology of linear maps, I completely prove the Jacobian conjecture by demonstrating the injectivity of real Keller map of any $n$-dimensions.

Some remarks on the Jacobian conjecture and polynomial endomorphisms

In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization, which is that under some conditions, a polynomial endomorphism with $r$ homogeneous parts of positive degree does not have $r$ times the same image point on a line through the origin, in case its Jacobian determinant does not vanish anywhere on that line. As a consequence, a Keller map of degree $r$ does not take the same values on $r > 1$ collinear points, provided $r$ is a unit in the base field. Next, we show that for invertible maps $x + H$ of deg…

Classification of cubic homogeneous polynomial maps with Jacobian matrices of rank two

Let $K$ be any field with $\textup{char}K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ with $\textup{rk} JH\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map then $F$ is invertible, and furthermore $F$ is tame if the dimension $n\neq 4$.

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.

Triangularization properties of power linear maps and the Structural Conjecture

In this paper, we discuss several additional properties a power linear Keller map may have. The Structural Conjecture by Druzkowski in [Dru] asserts that two such properties are equivalent, but we show that one of this properties is stronger than the other. We even show that the property of linear triangularizability is strictly in between. Furthermore, we give some positive results for small dimensions and small Jacobian ranks.