Related papers: On the Jacobian conjecture in characteristic zero

In his previous paper (Math. Res. Letters 7(2000), 123--132) the author proved that in characteristic zero the jacobian $J(C)$ of a hyperelliptic curve $C: y^2=f(x)$ has only trivial endomorphisms over an algebraic closure $K_a$ of the…

The real Jacobian conjecture was posed by Randall in 1983. This conjecture asserts that if $F=\left(f_1,\ldots ,f_n\right):\mathbb{R}^n\rightarrow\mathbb{R}^n$ is a polynomial map such that $\det DF\left(\mathbf{x}\right)\neq0$ for all…

Let $K$ be any field and $x = (x_1,x_2,\ldots,x_n)$. We classify all matrices $M \in {\rm Mat}_{m,n}(K[x])$ whose entries are polynomials of degree at most 1, for which ${\rm rk} M \le 2$. As a special case, we describe all such matrices…

In this paper, we study higher derivations of Jacobian type in positive characteristic. We give a necessary and sufficient condition for $(n-1)$-tuples of polynomials to be extendable in the polynomial ring in $n$ variables over an integral…

In this paper we show that the image of any locally finite $k$-derivation of the polynomial algebra $k[x, y]$ in two variables over a field $k$ of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian…

Let $K$ be a nonempty finite subset of the Euclidean space $\mathbb{R}^k$ $(k\ge 2)$. We prove that if a function $f\colon \mathbb{R}^k\to \mathbb{C}$ is such that the sum of $f$ on every congruent copy of $K$ is zero, then $f$ vanishes…

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

We prove that the Jacobian conjecture is false if and only if there exists a solution to a certain system of polynomial equations. We analyse the solution set of this system. In particular we prove that it is zero dimensional.

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…

We develop a cohomological description of various explicit descents in terms of generalized Jacobians, generalizing the known description for hyperelliptic curves. Specifically, given an integer $n$ dividing the degree of some reduced…

We propose a handful of definitions of injectivity for a parametrized family of maps and study its link with a global nonuniform stability conjecture for nonautonomous differential systems, which has been recently introduced. This relation…

Let K be an algebraically closed field of characteristic zero and let f(x,y) be a nonzero polynomial of K[x,y]. We prove that if the generic element of the family $(f-\lambda)\_{\lambda}$ is a rational polynomial, and if the Jacobian J(f,g)…

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…

Let $\Omega\subset \mathbb{R}^n$ be open and let $\mathcal{R}$ be a partial frame on $\Omega$, that is a set of $m$ linearly independent vector fields prescribed on $\Omega$ ($m\leq n$). We consider the issue of describing the set of all…

In this work we generalise the main result of arXiv:1812.05651 to the family of hyperelliptic curves with potentially good reduction over a $p$-adic field which have degree $p$ and the largest possible image of inertia under the $\ell$-adic…

For k=1,2,... infty and a Frolicher-Kriegl order k Lipschitz differentiable map f:E supseteq U to E having derivative at x_0 in U a linear homeomorphism E to E and satisfying a Colombeau type tameness condition, we prove that x_0 has a…

Let $C$ be a smooth projective curve of genus $g \ge 1$ over a finite field $\F$ of cardinality $q$. In this paper, we first study $\#\J_C$, the size of the Jacobian of $C$ over $\F$ in case that $\F(C)/\F(X)$ is a geometric Galois…

It is proved that for a Jacobi pair $(F,G)\in C[x,y]^2$, the Keller mapping: $(a,b)\mapsto(F(a,b),G(a,b))$ for $(a,b)\in C^2$, is injective. In particular, the $2$-dimensional Jacobi conjecture holds.

Consider the (formal/analytic/algebraic) map-germs Maps(X,(k^p,o)). Let G be the group of right/contact/left-right transformations. I extend the following (classical) results from the real/complex-analytic case to the case of arbitrary…

Let $\mathbb K$ be an algebraically closed field of characteristic zero, $\mathbb K[x, y]$ the polynonial ring in variables $x$, $y$ and let $W_2(\mathbb K)$ be the Lie algebra of all $\mathbb K$-derivations on $\mathbb K[x, y]$. A…