arXiv subjects
Publications and source records attributed to Mutsuo Oka.
To the best of the authors' knowledge, all previously known examples of $μ$-, $μ^*$-, link-, or ordinary Zariski pairs of surface singularities in $\mathbb{C}^3$ consist of (possibly weighted) Lê-Yomdin singularities. In this paper, we present an example of a $μ$-Zariski pair involving surface singularities that are not of Lê-Yomdin type.
We investigate surface singularities defined by weighted-Lê-Yomdin polynomials, with a particular focus on a specific subclass that we refer to as Newton weighted-Lê-Yomdin polynomials. In particular, using polynomials in this subclass, we develop a method to construct new $μ$-Zariski pairs of surface singularities.
Let $f$ be a (possibly Newton degenerate) weighted homogeneous polynomial defining an isolated surface singularity at the origin of $\mathbb{C}^3$, and let $\{f_s\}$ be a generic deformation of its coefficients such that $f_s$ is Newton non-degenerate for $s\not=0$. We show that there exists an ''admissible'' simultaneous good resolution of the family of functions $f_s$ for all small $s$, including $s=0$ which corresponds to the (possibly Newton degenerate) function $f$. As an application, we give a new geometrical proof of a weak version of the Milnor-Orlik theorem that asserts that the monodromy zeta-function of $f$ (and hence its Milnor number) is completely determined by its weight, its weighted degree and its Newton boundary.
We introduce a class of complex surface singularities - the blow-$ADE$ singularities - which are likely to be stable with respect to $μ^*$-constant deformations. We prove such a stability property in several special cases. Here, we emphasize that we are not just considering deformation families for small values of the deformation parameter but families connecting any two elements in the $μ^*$-constant stratum.
A Zariski pair of surfaces is a pair of complex polynomial functions in $\mathbb{C}^3$ which is obtained from a classical Zariski pair of projective curves $f_0(z_1,z_2,z_3)=0$ and $f_1(z_1,z_2,z_3)=0$ of degree $d$ in $\mathbb{P}^2$ by adding a same term of the form $z_i^{d+m}$ ($m\geq 1$) to both $f_0$ and $f_1$ so that the corresponding affine surfaces of $\mathbb{C}^3$ -- defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ -- have an isolated singularity at the origin and the same zeta-function for the monodromy associated with their Milnor fibrations (so, in particular, $g_0$ and $g_1$ have the same Milnor number). In the present paper, we show that if $f_0$ and $f_1$ are "convenient" with respect to the coordinates $(z_1,z_2,z_3)$ and if the singularities of the curves $f_0=0$ and $f_1=0$ are Newton non-degenerate in some suitable local coordinates, then $(g_0,g_1)$ is a $μ^*$-Zariski pair of surfaces, that is, a Zariski pair of surfaces whose polynomials $g_0$ and $g_1$ have the same Teissier's $μ^*$-sequence but lie in different path-connected components of the $μ^*$-constant stratum. To this end, we prove a new general formula that gives, under appropriate conditions, the Milnor number of functions of the above type, and we show (in a general setting) that two polynomials functions lying in the same path-connected component of the $μ^*$-constant stratum can always be joined by a "piecewise complex-analytic path".
The notion of Zariski pairs for projective curves in $\mathbb P^2$ is known since the pioneer paper of Zariski \cite{Zariski} and for further development, we refer the reference in \cite{Bartolo}.In this paper, we introduce a notion of Zariski pair of links in the class of isolated hypersurface singularities. Such a pair is canonically produced from a Zariski (or a weak Zariski ) pair of curves $C=\{f(x,y,z)=0\}$ and $C'=\{g(x,y,z)=0\}$ of degree $d$ by simply adding a monomial $z^{d+m}$ to $f$ and $g$ so that the corresponding affine hypersurfaces have isolated singularities at the origin. They have a same zeta function and a same Milnor number (\cite{Almost}). We give new examples of Zariski pairs which have the same $μ^*$ sequence and a same zeta function but two functions belong to different connected components of $μ$-constant strata (Theorem \ref{mu-zariski}). Two link 3-folds are not diffeomorphic and they are distinguished by the first homology which implies the Jordan form of their monodromies are different (Theorem \ref{main2}). We start from weak Zariski pairs of projective curves to construct new Zariski pairs of surfaces which have non-diffeomorphic link 3-folds. We also prove that hypersurface pair constructed from a Zariski pair give a diffeomorphic links (Theorem \ref{main3}).
It is well known that the diffeomorphism-type of the Milnor fibration of a (Newton) non-degenerate polynomial function $f$ is uniquely determined by the Newton boundary of $f$. In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism-type of the Milnor fibration of a (possibly degenerate) polynomial function of the form $f=f^1\cdots f^{k_0}$ is uniquely determined by the Newton boundaries of $f^1,\ldots, f^{k_0}$ if $\{f^{k_1}=\cdots=f^{k_m}=0\}$ is a non-degenerate complete intersection variety for any $k_1,\ldots,k_m\in \{1,\ldots, k_0\}$.
Let $f(\mathbf z)$ be an analytic function defined in the neighborhood of the origin of $\mathbb C^n$ which have some Newton degenerate faces. We generalize the Varchenko formula for the zeta function of the Milnor fibration of a Newton non-degenerate function $f$ to this case. As an application, we give an example of a pair of hypersurfaces with the same Newton boundary and the same zeta function with different tangent cones.
We consider the exponent of Łojasiewicz inequality $\|\partial\,f(\mathbf z)\| \ge c |f(\mathbf z|^θ$ for two classes of analytic functions and we will give an explicit estimation for $θ$. First we consider certain non-degenerate functions which is not convenient. In §3.4, we give an example of a polynomial for which $θ_0(f)$ is not constant on the moduli space and in §3.5, we show that the behaviors of the Łojasiewicz exponents is not similar as the Milnor numbers by an example. In the last section (§4), we give also an estimation for product functions $f(\mathbf z)=f_1(\mathbf z)\cdots f_k(\mathbf z)$ associated to a family of a certain convenient non-degenerate complete intersection varieties. In either class, the singularity is not isolated. We will give explicit estimations of the Łojasiewicz exponent $θ_0(f)$ using combinatorial data of the Newton boundary of $f$. We generalize this estimation for non-reduced function $g=f_1^{m_1}\cdots f_k^{m_k}$.
We give a criterion to test geometric properties such as Whitney equisingularity and Thom's $a_f$ condition for new families of (possibly non-isolated) hypersurface singularities that "behave well" with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.
We consider a mixed function of type $H(z,\bar z)=f(z)\bar g(z)$ where $f,g$ are non-degenerate but they are not assumed to be convenient. We assume that $f=0$ and $g=0$ and $f=g=0$ are non-degenerate and locally tame. We will show that $H$ has a tubular Milnor fibration and a spherical Milnor fibration. We show also two fibrations are equivalent.
We consider a mixed function of type $H(\mathbf z,\bar {\mathbf z})=f(\mathbf z)\bar g(\mathbf z)$ where $f$ and $g$ are convenient holomorphic functions which have isolated critical points at the origin and we assume that the intersection $f=g=0$ is a complete intersection variety with an isolated singlarity at theorigin. We assume also that $H$ satisfies the multiplicity condition.We will show that $H$ has a tubular Milnor fibration and also a spherical Milnor fibration. We give examples which does not satisfy the Newton multiplicity condition where one does not have Milnor fibration and the others have Milnor fibrations.
In this note, we prove the connectivity of the Milnor fiber for a mixed polynomial $f(\mathbf z,\bar{\mathbf z})$, assuming the existence of a sequence of smooth points of $f^{-1}(0)$ converging to the origin. This result gives also a another proof for the connectivity of the Milnor fiber of a non-reduced complex analytic function which is proved by A. Dimca
Let $f(\bf z,\bar{\bf z})$ be a strongly mixed homogeneous polynomial of 3 variables $\bf z=(z_1,z_2,z_3)$ of polar degree $q$ with an isolated singularity at the origin. It defines a smooth Riemann surface $C$ in the complex projective space $\mathbb P^2$. The fundamental group of the complement $\mathbb P^2\setminus C$ is cyclic group of order $q$ if $f$ is homogeneous polynomial without $\bar{\bf z}$. We propose a conjecture that this may be even true for mixed homogeneous polynomials by giving several supporting examples.
We consider roots of a generalized Lens polynomial $L(z,\bar z)={\bar z}^m q(z)-p(z)$ and also harmonically splitting Lens type polynomial $L^{hs}(z,\bar z)=r(\bar z)q(z)-p(z)$ and with ${\rm deg}\,q(z)=n$, ${\rm deg}\,r(\bar z)=m$ and ${\rm deg}\,p(z)\le n$. We have shown that there exists a harmonically splitting polynomial $r(\bar z)q(z)-p(z)$ which takes $5n+m-6$ roots, using a bifurcation family of polynomials. In this note, we show that this number can be taken by a generalized Lens polynomial ${\bar z}^mq(z)-p(z)$ after a slight modification of the bifurcation family of a Rhie polynomial.
We consider Łojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the Łojasiewicz exponent in a slightly weaker form than the assertion in Fukui.For a weighted homogeneous polynomial, we give a better estimation in the form which is conjectured by Brzostowski, Krasinski and Oleksik under under some condition (the Łojasiewicz non-degeneracy). We also introduce Łojasiewicz inequality for strongly non-degenerate mixed functions and generalize this estimation for mixed functions.
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 condition.
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.