arXiv subjects
Publications and source records attributed to Nguyen Thi Bich Thuy.
We describe a singular variety associated to the smallest degree Pinchuk map and calculate its intersection homology. The result describes the geometry at infinity of the Pinchuk's map.
The use of Cauchy's method to prove Euler's well-known formula is an object of many controversies. The purpose of this paper is to prove that Cauchy's method applies for convex polyhedra and not only for them, but also for surfaces such as the torus, the projective plane, the Klein bottle and the pinched torus.
The use of Cauchy's method in proving the well-known Euler formula is an object of many controversies. The purpose of this paper is to prove that the Cauchy's method applies for convex polyhedra and not only for them, but also for surfaces such as the torus, the projective plane, the Klein bottle and the pinched torus.
The Poincare-Hopf Theorem is one of the most used in other areas of science. There are applications of the Poincare-Hopf Theorem in physics, chemistry, biology and even in economics, psychology, etc ... The Poincare-Hopf Theorem connects an invariant of combinatorial, the character of Euler-Poincare to an invariant of differential geometry, index of vector fields. The results that connect two very different areas of mathematics can be considered as the most beautiful, useful and fruitful.
We introduce the Euler-Poincaré's characteristic with an elementary way and historically. We explain also why one should call Descartes-Poincaré characteristic instead of the Euler-Poincaré's characteristic. All the considered spaces are compact and without boundary. We work essentially on smooth and oriented surfaces . However, we work also on singular and non-oriented surfaces. We provide also the results which may be extended for the case of superior dimension.
We prove that the stratification of the asymptotic set associated to a polynomial mapping $F: \mathbb{C}^n \to \mathbb{C}^n$ defined by "la méthode des façons" (arXiv:1407.5329) is a Thom-Mather stratification. Nous prouvons que la stratification de lensemble asymptotique dune application polynomial mapping $F: \mathbb{C}^n \to \mathbb{C}^n$ définie par "la méthode des façons" (arXiv:1407.5329) est une stratification de Thom-Mather
Nous donnons une méthode naturelle, que nous appelons {\it "la méthode des façons"} pour stratifier la variété asymptotique associée à une application polynomiale dominante $F: \mathbb{C}^n \to \mathbb{C}^n$. La stratification obtenue est différentiable et satisfait la propriété de frontière.
We provide an algorithm to classify the asymptotic sets of the dominant polynomial mappings $F: \C^3 \to \C^3$ of degree 2, using the definition of the so-called "{\it façons}" in \cite{Thuy}. We obtain a classification theorem for the asymptotic sets of dominant polynomial mappings $F: \C^3 \to \C^3$ of degree 2. This algorithm can be generalized for the dominant polynomial mappings $F: \C^n \to \C^n$ of degree $d$, with any $(n, d) \in {(\N^*)}^2$.
In \cite{Valette}, Guillaume and Anna Valette associate singular varieties $V_F$ to a polynomial mapping $F: \C^n \to \C^n$. In the case $F: \C^2 \to \C^2$, if the set $K_0(F)$ of critical values of $F$ is empty, then $F$ is not proper if and only if the 2-dimensional homology or intersection homology (with any perversity) of $V_F$ are not trivial. In \cite{ThuyValette}, the results of \cite{Valette} are generalized in the case $F: \C^n \to \C^n$ where $n \geq 3$, with an additional condition. In this paper, we prove that if $F: \C^2 \to \C^2$ is a non-proper {\it generic dominant} polynomial mapping, then the 2-dimensional homology and intersection homology (with any perversity) of $V_F$ are not trivial. We prove that this result is true also for a non-proper {\it generic dominant} polynomial mapping $F: \C^n \to \C^n$ ($\, n \geq 3$), with the same additional condition than in \cite{ThuyValette}. In order to compute the intersection homology of the variety $V_F$, we provide an explicit Thom-Mather stratification of the set $K_0(F) \cup S_F$.
We construct a singular variety ${\mathcal{V}}_G$ associated to a polynomial mapping $G : \C^{n} \to \C^{n - 1}$ where $n \geq 2$. We prove that in the case $G : \C^{3} \to \C^{2}$, if $G$ is a local submersion but is not a fibration, then the homology and the intersection homology with total perversity (with compact supports or closed supports) in dimension two of the variety ${\mathcal{V}}_G$ is not trivial. In the case of a local submersion $G : \C^{n} \to \C^{n - 1}$ where $n \geq 4$, the result is still true with an additional condition.
In the paper "Geometry of polynomial mapping at infinity via intersection homology" the second and third authors associated to a given polynomial mapping $F : \C^2 \to \C^2$ with nonvanishing jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of the mapping. We generalize this result.