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Nguyen Van Chau

Publications and source records attributed to Nguyen Van Chau.

14 recordsLinked to original sources
arXiv2020

Jacobian conjecture as a problem on integral points on affine curves

It is shown that the $n$-dimensional Jacobian conjecture over algebraic number fields may be considered as an existence problem of integral points on affine curves. More specially, if the Jacobian conjecture over $\mathbb{C}$ is false, then for some $n\gg 1$ there exists a counterexample $F\in \mathbb{Z}[X]^n$ of the form $F_i(X)=X_i+ (a_{i1}X_1+\dots+a_{in}X_n)^{d_i}$, $a_{ij}\in \Z$, $d_i=2;3 $, $i,j=\overline{1,n},$ such that the affine curve $F_1(X)=F_2(X)=\dots=F_n(X)$ has no non-zero integer points.

math.AG↗
arXiv2017

Integral points on plane curves and Plane Jacobian Conjecture over number fields

Let $K$ be a number field and $O_K$ the ring of integers of $K$. In the spirit of Siegel's theorem on integral points on affine algebraic curves, the plane Jacobian conjecture over $K$ is equivalent to the following statement: if $P,Q\in O_K[x,y]$ and $P_xQ_y-P_yQ_x\equiv 1$, then the curve $P=0$ has more than one integral point.

math.AG↗
arXiv2010

A note on the plane Jacobian conjecture

It is shown that every polynomial function $P : \mathbb{C}^2\longrightarrow \mathbb{C}$ with irreducible fibres of same a genus is a coordinate. In consequence, there does not exist counterexamples F = (P,Q) to the Jacobian conjecture such that all fibres of P are irreducible curves of same a genus.

math.AG↗
arXiv2010

Pencil of irreducible rational curves and Plane Jacobian conjecture

We are concerned with the behavior of the polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ with finite fibres and satisfying the condition that all of the curves $aP+bQ=0$, $(a:b)\in \mathbb{P}^1$, are irreducible rational curves. The obtained result shows that such polynomial maps $F$ is invertible if $(0,0)$ is a regular value of $F$ or if the Jacobian condition holds.

math.AG↗
arXiv2007

Plane Jacobian conjecture for simple polynomials

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 $p:X\longrightarrow \mathbb{P}^1$ of a compactification $X$ of $\mathbb{C}^2$, the restriction of $p$ to each irreducible component $C$ of the compactification divisor $D = X-\mathbb{C}^2$ is either degree 0 or 1.

math.AG↗
arXiv2005

Iterated Images and the Plane Jacobian Conjecture

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 generally, we obtain the same result for open polynomial maps of a closed algebraic subset X of complex N-space to itself that have finite coimage, and for cofinite subsets of such an X invariant under the map. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity.

math.AG↗
arXiv2004

A Simple Proof of Jung's Theorem on Polynomial Automorphisms of $\C^2$

The Automorphism Theorem, discovered first by Jung in 1942, asserts that if $k$ is a field, then every polynomial automorphism of $k^2$ is a finite product of linear automorphisms and automorphisms of the form $(x,y)\mapsto(x+p(y), y) $ for $p\in k[y]$. We present here a simple proof for the case $k=\C$ by using Newton-Puiseux expansions.

math.AG↗