Related papers: New sufficient condition for the two-dimensional r…

This paper develops our previous work on properness of a class of maps related to the Jacobian conjecture. The paper has two main parts: - In part 1, we explore properties of the set of non-proper values $S_f$ (as introduced by Z. Jelonek)…

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…

Let $n\geq 2$ and $\mathbb K $ be a number field of characteristic $0$. Jacobian Conjecture asserts for a polynomial map $\mathcal P$ from $\mathbb K ^n$ to itself, if the determinant of its Jacobian matrix is a nonzero constant in $\mathbb…

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.

We show that the Jacobian conjecture of the two dimensional case is true.

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…

This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such…

Let $(F,G)$ be a Jacobian pair with $d=w\text{-deg}(F)$ and $e=w\text{-deg}(G)$ for some direction $w$. A generalized Magnus' formula approximates $G$ as $\sum_{\gamma\ge 0} c_\gamma F^{\frac{e-\gamma}{d}}$ for some complex numbers…

We prove a sufficient condition for the Jacobian problem in the setting of real, complex and mixed polynomial mappings. This follows from the study of the bifurcation locus of a mapping subject to a new Newton non-degeneracy condition.

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…

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)…

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

In this paper, we show that the Jacobian conjecture holds for gradient maps in dimension n <= 3 over a field K of characteristic zero. We do this by extending the following result for n <= 2 by F. Dillen to n <= 3: if f is a polynomial of…

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…

We prove that a polynomial map is invertible if and only if some associated differential ring homomorphism is bijective. To this end, we use a theorem of Crespo and Hajto linking the invertibility of polynomial maps with Picard-Vessiot…

We introduce a new non-degeneracy condition at infinity for a real or a mixed polynomial mapping $F$ which allows us to approximate its bifurcation locus in terms of certain Newton polyhedra. We derive a sufficiency result for the Jacobian…

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…

The direct or algorithmic approach for the Jacobian problem, consisting of the direct construction of the inverse polynomials is proposed. The so called principle and derived Jacobi conditions are proposed and discussed. The algorithmic…

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…

Using the local bijectivity of Keller maps, we give a proof of two-dimensional Jacobian conjecture.