Related papers: A characterization of Keller maps

Let K be an algebraically closed field of characteristic zero. We say that a polynomial automorphism f : K^n -> K^n is special if the Jacobian of f is equal to 1. We show that every (n - 1)-dimensional component H of the set Fix(f) of fixed…

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,…

We present a generalization of the Jacobian Conjecture for m polynomials in n variables: f1,...,fm belonging to k[x1,...,xn], where k is a field of characteristic zero and m=1,...,n. We express the generalized Jacobian condition in terms of…

The two-dimensional Jacobian Conjecture says that a Keller map $f: (x,y) \mapsto (p,q) \in k[x,y]^2$ having an invertible Jacobian is an automorphism of $k[x,y]$. We prove that there is no Keller map with $[k(x,y): k(p,q)]$ prime.

The Jacobian Conjecture uses the equation $det(Jac(F))\in k^*$, which is a very short way to write down many equations putting restrictions on the coefficients of a polynomial map $F$. In characteristic $p$ these equations do not suffice to…

For K a field of characteristic 0 and d any integer number greater than or equal to 2, we prove the invertibility of polynomial endomorphisms of the affine space of dimension d over K of the form F=Id+H, where each coordinate of H is the…

Let $K=k(C)$ be the function field of a complete nonsingular curve $C$ over an arbitrary field $k$. The main result of this paper states that a morphism $\phi:{\mathbb P}^N_K\to{\mathbb P}^N_K$ is isotrivial if and only if it has potential…

Let $k$ be an algebraically closed field of characteristic zero. Let $H:k^2\to k^2$ be a polynomial mapping such that the Jacobian $\text{Jac}\,H$ is a non-zero constant. In this note we prove, that if there is a line $l \subset k^2$ such…

Our goal is to settle the following faded problem: The Jacobian Conjecture (JC_n): If f_1,..,f_n are elements in a polynomial ring k[X_1,..,X_n] over a field k of characteristic 0 such that det(\partial f_i/ \partial X_j) is a nonzero…

Using the Galois theory over function field, and the holomorphy of algebroids defined via irreducible polynomial at singular points, we prove the injectivity of any kellerian mapping. The famous Jacobian conjecture is true.

In this note, we propose a super version of Jacobian conjecture on the automorphisms of affine superspaces over an algebraically closed field $\mathbb{F}$ of characteristic $0$, which predicts that for a homomorphism $\varphi$ of the…

We describe the automorphism group of the endomorphism semigroup $\End(K[x_1,...,x_n])$ of ring $K[x_1,...,x_n]$ of polynomials over an {\it arbitrary} field $K$. A similar result is obtained for automorphism group of the category of…

We prove that an irreducible polynomial derivation in positive characteristic is a Jacobian derivation if and only if there exists an n-1-element p-basis of its ring of constants. In the case of two variables we characterize these…

Let $K$ be a field of characteristic $0$, and let $k \geq 2$ be an integer. We prove that every $K$-linear bijection $f \colon K[X] \to K[X]$ strongly preserving the set of $k$-free polynomials (or the set of polynomials with a $k$-fold…

Let $G$ be a finite group acting effectively on the complex affine plane. If the $G$-action commutes with an \'etale endomorphism $f$ of the affine plane and the order of $G$ is even then the endomorphism $f$ is an automorphism.

We classify all quadratic homogeneous polynomial maps $H$ and Keller maps of the form $x + H$, for which $rk J H = 3$, over a field $K$ of arbitrary characteristic. In particular, we show that such a Keller map (up to a square part if $char…

In this paper we present a theorem concerning an equivalent statement of the Jacobian Conjecture in terms of Picard-Vessiot extensions. Our theorem completes the earlier work of T. Crespo and Z. Hajto which suggested an effective criterion…

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…

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 $p:X\longrightarrow \mathbb{P}^1$…

In this paper, we will first show that, the homogeneous polynomials which satisfy the Jacobian condition are injective on the lines that pass through the origin. Secondly, we will show that $F$ and $G'$ are paired, where $F$ is a Druzkowski…