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Sep 19, 2013
09/13

by
Mutsuo Oka

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Mixed functions are analytic functions in variables $z_1,..., z_n$ and their conjugates $\bar z_1,..., \bar z_n$. We introduce the notion of Newton non-degeneracy for mixed functions and develop a basic tool for the study of mixed hypersurface singularities. We show the existence of a canonical resolution of the singularity, and the existence of the Milnor fibration under the strong non-degeneracy condition.

Source: http://arxiv.org/abs/0909.1904v1

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51

Jul 19, 2013
07/13

by
Mutsuo Oka

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Let $f(\bfz,\bar\bfz)$ be a mixed strongly polar homogeneous polynomial of $3$ variables $\bfz=(z_1,z_2, z_3)$. It defines a Riemann surface $V:=\{[\bfz]\in \BP^{2}\,|\,f(\bfz,\bar\bfz)=0 \}$ in the complex projective space $\BP^{2}$. We will show that for an arbitrary given $g\ge 0$, there exists a mixed polar homogeneous polynomial with polar degree 1 which defines a projective surface of genus $g$. For the construction, we introduce a new type of weighted homogeneous polynomials which we...

Source: http://arxiv.org/abs/1005.1449v1

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44

Sep 22, 2013
09/13

by
Mutsuo Oka

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Alexander polynomials of sextics with only simple singularities or sextics of torus type with arbitrary singularities are computed. We show that for ieeducible sextics,there are four possibilities: $(t^2-t+1)^j, j=0,1,2,3$.

Source: http://arxiv.org/abs/math/0205092v1

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Sep 19, 2013
09/13

by
Mutsuo Oka

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Let $f_{{\bf a},\{bf b}}({\bf z},\bar{\bf z})=z_1^{a_1+b_1}\bar z_1^{b_1}+...+z_n^{a_n+b_n}\bar z_n^{b_n}$ be a polar weighted homogeneous mixed polynomial with $a_j>0,b_j\ge 0$, $j=1,..., n$ and let $f_{{\bf a}}({\bf z})=z_1^{a_1}+...+z_n^{a_n}$ be the associated weighted homogeneous polynomial. Consider the corresponding link variety $K_{{\bf a},{\bf b}}=f_{{\bf a},{\bf b}}\inv(0)\cap S^{2n-1}$ and $K_{{\bf a}}=f_{{\bf a}}\inv(0)\cap S^{2n-1}$. Ruas-Seade-Verjovsky \cite{R-S-V} proved that...

Source: http://arxiv.org/abs/0909.4605v2

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Sep 21, 2013
09/13

by
Mutsuo Oka

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We give a complete classsification of reduced sextics of torus type with configurations of the singularities and the geometry of the components.

Source: http://arxiv.org/abs/math/0203034v1

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2.0

Jun 30, 2018
06/18

by
Mutsuo Oka

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Convenient mixed functions with strongly non-degenerate Newton boundaries have Milnor fibrations, as the isolatedness of the singularity follows from the convenience. In this paper, we consider the Milnor fibration for non-convenient mixed functions. We also study geometric properties such as Thom's $a_f$ condition, the transversality of the nearby fibers and stable boundary property of the Milnor fibration and their relations.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1409.5275

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Sep 19, 2013
09/13

by
Mutsuo Oka

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The complete list of reducible sextics of torus type with simple singularities is known in our previous paper. In this paper, we give a complete list of existence and non-existence of Zariski partner sextics of non-torus type corresponding to the list of sextics of torus type.

Source: http://arxiv.org/abs/math/0507052v1

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Sep 21, 2013
09/13

by
Mutsuo Oka

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We consider two mixed curve $C,C'\subset {\Bbb C}^2$ which are defined by mixed functions of two variables $\bf z=(z_1,z_2)$. We have shown in \cite{MC}, that they have canonical orientations. If $C$ and $C'$ are smooth and intersect transversely at $P$, the intersection number $I_{top}(C,C';P)$ is topologically defined. We will generalize this definition to the case when the intersection is not necessarily transversal or either $C$ or $C'$ may be singular at $P$ using the defining mixed...

Source: http://arxiv.org/abs/1104.3386v1

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Sep 19, 2013
09/13

by
Mutsuo Oka

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Let $\mathcal N$ be the moduli space of sextics with 3 (3,4)-cusps. The quotient moduli space ${\mathcal N}/G$ is one-dimensional and consists of two components, ${\mathcal N}_{torus}/G$ and ${\mathcal N}_{gen}/G$. By quadratic transformations, they are transformed into one-parameter families $C_s$ and $D_s$ of cubic curves respectively. We study the Mordell-Weil torsion groups of cubic curves $C_s$ over $\bfQ$ and $D_s$ over $\bfQ(\sqrt{-3})$ respectively. We show that $C_{s}$ has the torsion...

Source: http://arxiv.org/abs/math/9912041v3

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Sep 19, 2013
09/13

by
Mutsuo Oka

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Let $f(\bfz,\bar\bfz)$ be a mixed polar homogeneous polynomial of $n$ variables $\bfz=(z_1,..., z_n)$. It defines a projective real algebraic variety $V:=\{[\bfz]\in \BC\BP^{n-1} | f(\bfz,\bar\bfz)=0 \}$ in the projective space $\BC\BP^{n-1}$. The behavior is different from that of the projective hypersurface. The topology is not uniquely determined by the degree of the variety even if $V$ is non-singular. We study a basic property of such a variety.

Source: http://arxiv.org/abs/0910.2523v1

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Sep 20, 2013
09/13

by
Mutsuo Oka

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We introduce a notion of tangential Alexander polynomials for plane curves and study the relation with $\theta$^Alexander polynomial. As an application, we use these polynomials to study a non-reduced degeneration $C_t \to D_0+jL$. We show that there exists a certain surjectivity of the fundamental groups and divisibility among their Alexander polynomials.

Source: http://arxiv.org/abs/math/0601236v2

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4.0

Jun 27, 2018
06/18

by
Mutsuo Oka

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We consider a certain mixed polynomial which is an extended Lens equation $L_{n,m}=\bar z^m-p(z)/q(z)$ with $\text{degree}\, q=n$, $\text{degree}\, p

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1505.03576

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Sep 21, 2013
09/13

by
Mutsuo Oka

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A strongly non-degenerate mixed function has a Milnor open book structures on a sufficiently small sphere. We introduce the notion of {\em a holomorphic-like} mixed function and we will show that a link defined by such a mixed function has a canonical contact structure. Then we will show that this contact structure for a certain holomorphic-like mixed function is carried by the Milnor open book.

Source: http://arxiv.org/abs/1204.5528v1

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Sep 19, 2013
09/13

by
Mutsuo Oka

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We show the existence of sextics of non-torus type which is a Zariski partner of the tame sextics of torus type with simple singularities.

Source: http://arxiv.org/abs/math/0507051v1

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Sep 23, 2013
09/13

by
Mutsuo Oka

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Let $f(\bf z,\bar{\bf z})$ be a mixed polynomial with strongly non-degenerate face functions. We consider a canonical toric modification $\pi:\,X\to \Bbb C^n$ and a polar modification $\pi_{\Bbb R}:Y\to X$. We will show that the toric modification resolves topologically the singularity of $V$ and the zeta function of the Milnor fibration of $f$ is described by a formula of a Varchenko type.

Source: http://arxiv.org/abs/1202.2166v1

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Sep 23, 2013
09/13

by
Mutsuo Oka

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Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex weighted homogeneous polynomials. We study basic properties of hypersurfaces which are defined by polar weighted homogeneous polynomials.

Source: http://arxiv.org/abs/0801.3708v1

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3.0

Jun 29, 2018
06/18

by
Christophe Eyral; Mutsuo Oka

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We investigate the equisingularity question for $1$-parameter deformation families of mixed polynomial functions $f_t(\mathbf{z},\bar{\mathbf{z}})$ from the Newton polygon point of view. We show that if the members $f_t$ of the family satisfy a number of elementary conditions, which can be easily described in terms of the Newton polygon, then the corresponding family of mixed hypersurfaces $f_t^{-1}(0)$ is Whitney equisingular (and hence topologically equisingular) and satisfies the Thom...

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1607.03741

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7.0

Jun 26, 2018
06/18

by
Vincent Blanlœil; Mutsuo Oka

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We consider a canonical $S^1$ action on $S^3$ which is defined by $(\rho,(z_1,z_2))\mapsto (z_1\rho^p,z_2\rho^q)$ for $\rho\in S^1$ and $(z_1,z_2)\in S^3\subset {\mathbb C}^2$. We consider a link consisting of finite orbits of this action, which some of the orbits are reversely oriented. Such a link appears as a link of a certain type of mixed polynomials. We study the space of such links and show smooth degeneration relations.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1501.05106

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43

Sep 18, 2013
09/13

by
Guangfeng Jiang; Mutsuo Oka

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On an affine variety $X$ defined by homogeneous polynomials, every line in the tangent cone of $X$ is a subvariety of $X$. However there are many other germs of analytic varieties which are not of cone type but contain ``lines'' passing through the origin. In this paper, we give a method to determine the existence and the ``number'' of such lines on non-degenerate surface singualrities.

Source: http://arxiv.org/abs/math/0005070v1

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Sep 17, 2013
09/13

by
Alex Degtyarev; Mutsuo Oka

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We analyze irreducible plane sextics whose fundamental group factors to $D_{14}$. We produce explicit equations for all curves and show that, in the simplest case of the set of singularities $3A_6$, the group is $D_{14}\times Z_3$.

Source: http://arxiv.org/abs/0711.3067v1

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Sep 22, 2013
09/13

by
Christophe Eyral; Mutsuo Oka

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The existence of Alexander-equivalent Zariski pairs dealing with irreducible curves of degree 6 was proved by A. Degtyarev. However, up to now, no explicit example of such a pair was available (only the existence was known). In this paper, we construct the first concrete example.

Source: http://arxiv.org/abs/0811.2310v1

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Sep 21, 2013
09/13

by
Norbert A'Campo; Mutsuo Oka

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We show the existence of toric resolution tower for an irreducible curve singularity which is explicitly described by Tschirnhausen polynomials. We deduce for a smooth affine plane curve from its topology restrictions for its singularity at infinity. For instance, we discribe the singularities at infinity (up to equisingular deformation) for curves of genus 0,1 and 2. From this follows an extension of the Abhyankar-Moh-Suzuki theorem to genus 1 and 2.

Source: http://arxiv.org/abs/alg-geom/9508007v2

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2.0

Jun 28, 2018
06/18

by
Christophe Eyral; Mutsuo Oka

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In an unpublished lecture note, J. Brian\c{c}on observed that if $\{f_t\}$ is a family of isolated complex hypersurface singularities such that the Newton boundary of $f_t$ is independent of $t$ and $f_t$ is non-degenerate, then the corresponding family of hypersurfaces $\{f_t^{-1}(0)\}$ is Whitney equisingular (and hence topologically equisingular). A first generalization of this assertion to families with non-isolated singularities was given by the second author under a rather technical...

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1512.04248

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Sep 19, 2013
09/13

by
Mutsuo Oka; Duc Tai Pho

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We show that the fundamental group of the complement of any irreducible tame torus sextics in $\bf P^2$ is isomorphic to $\bf Z_2*\bf Z_3$ except one class. The exceptional class has the configuration of the singularities $\{C_{3,9},3A_2\}$ and the fundamental group is bigger than $\bf Z_2*\bf Z_3$. In fact, the Alexander polynomial is given by $(t^2-t+1)^2$. For the proof, we first reduce the assertion to maximal curves and then we compute the fundamental groups for maximal tame torus curves.

Source: http://arxiv.org/abs/math/0010182v1

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Sep 22, 2013
09/13

by
Mutsuo Oka; Duc Tai Pho

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The second author classified configurations of the singularities on tame sextics of torus type. In this paper, we give a complete classification of the singularities on irreducible sextics of torus type, without assuming the tameness of the sextics. We show that there exists 121 configurations and there are 5 pairs and a triple of configurations for which the corresponding moduli spaces coincide, ignoring the respective torus decomposition.

Source: http://arxiv.org/abs/math/0201035v1

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2.0

Jun 29, 2018
06/18

by
Kazumasa Inaba; Masayuki Kawashima; Mutsuo Oka

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We study a simplicial mixed polynomial of cyclic type and its associated weighted homogeneous polynomial. In the present paper, we show that their links are diffeomorphic and their Milnor fibrations are isomorphic.

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1606.03604