Related papers: Jacobian conjecture as a problem on integral point…
One of the aims of this article is to provide a class of polynomial mappings for which the Jacobian conjecture is true. Also, we state and prove several global univalence theorems and present a couple of applications of them.
We study the Jacobian scheme of a plane algebraic curve at an ordinary singularity, characterizing it through a geometric property. We compute the Tjurina number for a family of curves at an ordinary singularity showing that it reaches the…
By using techniques of holomorphic jets and Jacobian fields, we devise a non-equidistribution theory of holomorphic curves into complex projective varieties intersecting normal crossing divisors. Based on this theory established, we prove…
We show that Jacobian algebras arising from a sphere with $n$-punctures, with $n\geq5$, are finite dimensional algebras. We consider also a family of cyclically oriented quivers and we prove that, for any primitive potential, the associated…
Let us denote by $\mathcal K_n$ the hyperspace of all convex bodies of $\mathbb R^n$ equipped with the Hausdorff distance topology. An affine invariant point $p$ is a continuous and Aff(n)-equivariant map $p:\mathcal K_n\to \mathbb R^n$,…
We show that any polarized abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective version of Poonen's Bertini theorem over finite fields,…
In this paper we give a criterion of irreducibility for a complex power series in two variables, using the notion of jacobian Newton diagrams, defined with respect to any direction. Moreover we study the singularity at infinity of a plane…
We show that the iterated images of a Jacobian pair stabilize; that is, the k-th iterates of a polynomial map of complex two-space to itself with a nonzero constant Jacobian determinant all have the same image for sufficiently large k. More…
We examine \'etale covers of genus two curves that occur in the linear system of a polarizing line bundle of type $(1,d)$ on a complex abelian surface. We give results counting fixed points of involutions on such curves as well as…
Implementations of known reductions of the Strong Real Jacobian Conjecture (SRJC), to the case of an identity map plus cubic homogeneous or cubic linear terms, and to the case of gradient maps, are shown to preserve significant algebraic…
It is shown that a polynomial map $(P,Q)\in \mathbb{Q}[x,y]^2$ with $P_xQ_y-P_yQ_x \equiv 1$ has an inverse map in $\mathbb{Q}[x,y]^2$ if the fiber $P=0$ contains an infinite subset of $ d^{-1}\mathbb{Z}^2$ for an integer $d$.
The famous Jacobian problem asks: Is a morphism $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ having an invertible Jacobian, invertible? If we add the assumption that $\mathbb{C}(f(x),f(y))=\mathbb{C}(x,y)$, then $f$ is invertible; this result is…
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,…
Piontkowski calculated the Euler number of Jacobi factors of plane curve singularities with semigroups $< p, q>$, $< 4, 2q, s>$, $< 6,8,s>$ and $< 6,10, s>$. %His analysis was done by decomposing the Jacobi factors into affine cells. In…
We present some motivations and discuss various aspects of an approach to the Jacobian Conjecture in terms of irreducible elements and square-free elements.
Given an abelian variety $X$ and a point $a\in X$ we denote by $<a>$ the closure of the subgroup of $X$ generated by $a$. Let $N=2^g-1$. We denote by $\kappa: X\to \kappa(X)\subset\mathbb P^N$ the map from $X$ to its Kummer variety. We…
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 consider a pointed curve $(X,P)$ which is given by the Weierstrass normal form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\cdots + A_{r-1}(x) y + A_{r}(x)$ where $x$ is an affine coordinate on $\mathbb{P}^1$, the point $\infty$ on $X$…
We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of F-rational points. In the case of the Jacobian, we show that any…
In [5], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively in [1] to study the algebraic curves for…