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Pham Tien Son

Bio: Pham Tien Son is an academic researcher. The author has contributed to research in topics: Automorphism & Euler characteristic. The author has an hindex of 1, co-authored 1 publications receiving 4 citations.

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TL;DR: In this paper, a global version of the Le-Ramanujam μ-constant theorem for polynomials is proposed, where the Euler characteristic of a generic fiber of a polynomial is constant and the global monodromy fibrations of the fiber are all isomorphic.
Abstract: We are interested in a global version of Le-Ramanujam μ-constant theorem for polynomials. We consider an analytic family {fs}, s $\in$ [0, 1], of complex polynomials in two variables, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber of fs is constant, then we show that the global monodromy fibrations of fs are all isomorphic, and that the degree of fs is constant (up to an algebraic automorphism of C2).

4 citations


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TL;DR: In this paper, it was shown that the Milnor number of a non-degenerate isolated complete intersection singularity is invariant to the Newton polyhedra of the component functions.
Abstract: We prove that for two germs of analytic mappings $f,g\colon (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^p,0)$ with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family $\{f_t\}$ of analytic maps with $f_0=f, f_1=g$ which has a so-called {\it uniform stable radius for the Milnor fibration}. As a corollary, we show that their Milnor numbers are equal. Also, a formula for the Milnor number is given in terms of the Newton polyhedra of the component functions. This is a generalization of the result by C. Bivia-Ausina. Consequently, we obtain that the Milnor number of a non-degenerate isolated complete intersection singularity is an invariance of Newton boundaries.

2 citations

Posted Content
TL;DR: In this article, the authors give an explicit description of a finite set $T_\infty(f|_S) \subset \mathbb{C}$ such that the set of critical values of certain polynomial functions contained in such a set are all isomorphic under Newton non-degenerate at infinity.
Abstract: Let $S\subset \mathbb{C}^n$ be a non-singular algebraic set and $f \colon \mathbb{C}^n \to \mathbb{C}$ be a polynomial function. It is well-known that the restriction $f|_S \colon S \to \mathbb{C}$ of $f$ on $S$ is a locally trivial fibration outside a finite set $B(f|_S) \subset \mathbb{C}.$ In this paper, we give an explicit description of a finite set $T_\infty(f|_S) \subset \mathbb{C}$ such that $B(f|_S) \subset K_0(f|_S) \cup T_\infty(f|_S),$ where $K_0(f|_S)$ denotes the set of critical values of the $f|_S.$ Furthermore, $T_\infty(f|_S)$ is contained in the set of critical values of certain polynomial functions provided that the $f|_S$ is Newton non-degenerate at infinity. Using these facts, we show that if $\{f_t\}_{t \in [0, 1]}$ is a family of polynomials such that the Newton polyhedron at infinity of $f_t$ is independent of $t$ and the $f_t|_S$ is Newton non-degenerate at infinity, then the global monodromies of the $f_t|_S$ are all isomorphic.

2 citations

Posted Content
23 Dec 2019
TL;DR: In this paper, it was shown that the Milnor fibrations over a same base of a family of Newton nondegenerate isolated singularity complete intersections which have the same Newton boundaries are isomorphic.
Abstract: We prove that the Milnor fibrations over a same base of a family of Newton nondegenerate isolated singularity complete intersections which have the same Newton boundaries are isomorphic. As a consequence, we obtain that the Milnor number of a Newton nondegenerate complete intersection is an invariance of Newton boundaries.
Journal ArticleDOI
TL;DR: In this paper, the authors showed that the Milnor number of a non-degenerate isolated complete intersection singularity is invariant to the Newton polyhedra of the component functions.
Abstract: We prove that for two germs of analytic mappings $$f,g:({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}}^p,0)$$ with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family $$\{f_t\}$$ of analytic maps with $$f_0=f, f_1=g$$ which has a so-called uniform stable radius for the Milnor fibration. As a corollary, we show that their Milnor numbers are equal. Also, a formula for the Milnor number is given in terms of the Newton polyhedra of the component functions. This is a generalization of the result by C. Bivia-Ausina. Consequently, we obtain that the Milnor number of a non-degenerate isolated complete intersection singularity is an invariant of Newton boundaries.