Let f(š³) be an analytic function defined in a neighborhood of the origin of ā^n which have some N... more Let f(š³) be an analytic function defined in a neighborhood of the origin of ā^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 pair of hypersurfaces with the same Newton boundary and same zeta function with non-homeomorphic link manifolds.
We consider a mixed function of type H(z,zĢ )=f(z)gĢ (z) where f,g are non-degenerate but they are ... more We consider a mixed function of type H(z,zĢ )=f(z)gĢ (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 vibration and a spherical Milnor fibration. We show also two vibrations are equivalent.
The existence of Alexander-equivalent Zariski pairs dealing with irreducible curves of degree 6 w... more 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.
We show the existence of sextics of non-torus type which is a Zariski partner of the tame sextics... more 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.
An R-join-type curve is a curve in C^2 defined by an equation of the form aĀ·ā_j=1^ā (y-Ī²_j)^Ī½_j =... more An R-join-type curve is a curve in C^2 defined by an equation of the form aĀ·ā_j=1^ā (y-Ī²_j)^Ī½_j = bĀ·ā_i=1^m (x-Ī±_i)^Ī»_i, where the coefficients a, b, Ī±_i and Ī²_j are real numbers. For generic values of a and b, the singular locus of the curve consists of the points (Ī±_i,Ī²_j) with Ī»_i,Ī½_jā„ 2 (so-called inner singularities). In the non-generic case, the inner singularities are not the only ones: the curve may also have `outer' singularities. The fundamental groups of (the complements of) curves having only inner singularities are considered in O. In the present paper, we investigate the fundamental groups of a special class of curves possessing outer singularities.
We consider \L ojasiewicz inequalities for a non-degenerate holomorphic function with an isolated... more We consider \L ojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the \L ojasiewicz 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 \L ojasiewicz non-degeneracy). We also introduce \L ojasiewicz inequality for strongly non-degenerate mixed functions and generalize this estimation for mixed functions.
The second author classified configurations of the singularities on tame sextics of torus type. I... more 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.
We give a criterion to test geometric properties such as Whitney equisingularity and Thom's a... more 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 two mixed curve C,C'ā C^2 which are defined by mixed functions of two variables z... more We consider two mixed curve C,C'ā C^2 which are defined by mixed functions of two variables z=(z_1,z_2). We have shown in 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 polynomials.
Mixed functions are analytic functions in variables z_1,..., z_n and their conjugates zĢ _1,..., z... more Mixed functions are analytic functions in variables z_1,..., z_n and their conjugates zĢ _1,..., zĢ _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.
We consider a mixed function of type H(z,zĢ )=f(z)gĢ (z) where f and g are convenient holomorphic f... more We consider a mixed function of type H(z,zĢ )=f(z)gĢ (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.
Let f(,) be a mixed strongly polar homogeneous polynomial of 3 variables =(z_1,z_2, z_3). It defi... more Let f(,) be a mixed strongly polar homogeneous polynomial of 3 variables =(z_1,z_2, z_3). It defines a Riemann surface V:={[]ā^2 | f(,)=0 } in the complex projective space ^2. We will show that for an arbitrary given g> 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 call polar weighted homogeneous polynomials of twisted join type.
where z = z1 1 Ā· Ā· Ā· z Ī½n n for Ī½ = (Ī½1, . . . , Ī½n) (respectively zĢ Ī¼ = zĢ1 1 Ā· Ā· Ā· zĢ Ī¼n n for... more where z = z1 1 Ā· Ā· Ā· z Ī½n n for Ī½ = (Ī½1, . . . , Ī½n) (respectively zĢ Ī¼ = zĢ1 1 Ā· Ā· Ā· zĢ Ī¼n n for Ī¼ = (Ī¼1, . . . , Ī¼n)) as usual. Here zĢj is the complex conjugate of zj. We call f(z, zĢ) a mixed analytic function (or a mixed polynomial, if f(z, zĢ) is a polynomial) of z1, . . . , zn. We are interested in the topology of the hypersurface V = {z ā C | f(z, zĢ) = 0}, which we call a mixed hypersurface. This approach is equivalent to the original one.
J(f, g) := fx(x, y)gy(x, y) ā fy(x, y)gx(x, y) where fx, fy are partial derivatives with respect ... more J(f, g) := fx(x, y)gy(x, y) ā fy(x, y)gx(x, y) where fx, fy are partial derivatives with respect to x and y respectively. We can also understand J(f, g) by (df ā§ dg)(x, y) = J(f, g) dx ā§ dy. Suppose that the mapping Ī¦ : C ā C is an automorphism in the sense that it has an inverse polynomial mapping ĪØ : C ā C. Then by the composition rule of the Jacobian, we have the equality J(Ī¦) ā” c for a non-zero constant c. The Jacobian conjecture is concerned with the converse assertion: (JC): If J(f, g) ā” c, c ā Cā, (f, g) : C ā C is an automorphism. The first essential contribution to this conjecture was made by Suzuki [4], Abhyankar [1] and the author did a small work about outside faces of mixed weights using Newton diagram [3]. There are many papers since then about this topic but it seems that there are few further progress on this conjecture of two variable case. This note is a rewritten version of [3] for a conference talk in Hanoi, October 2006. We tried to make some argument simpler th...
Let f(š³) be an analytic function defined in a neighborhood of the origin of ā^n which have some N... more Let f(š³) be an analytic function defined in a neighborhood of the origin of ā^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 pair of hypersurfaces with the same Newton boundary and same zeta function with non-homeomorphic link manifolds.
We consider a mixed function of type H(z,zĢ )=f(z)gĢ (z) where f,g are non-degenerate but they are ... more We consider a mixed function of type H(z,zĢ )=f(z)gĢ (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 vibration and a spherical Milnor fibration. We show also two vibrations are equivalent.
The existence of Alexander-equivalent Zariski pairs dealing with irreducible curves of degree 6 w... more 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.
We show the existence of sextics of non-torus type which is a Zariski partner of the tame sextics... more 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.
An R-join-type curve is a curve in C^2 defined by an equation of the form aĀ·ā_j=1^ā (y-Ī²_j)^Ī½_j =... more An R-join-type curve is a curve in C^2 defined by an equation of the form aĀ·ā_j=1^ā (y-Ī²_j)^Ī½_j = bĀ·ā_i=1^m (x-Ī±_i)^Ī»_i, where the coefficients a, b, Ī±_i and Ī²_j are real numbers. For generic values of a and b, the singular locus of the curve consists of the points (Ī±_i,Ī²_j) with Ī»_i,Ī½_jā„ 2 (so-called inner singularities). In the non-generic case, the inner singularities are not the only ones: the curve may also have `outer' singularities. The fundamental groups of (the complements of) curves having only inner singularities are considered in O. In the present paper, we investigate the fundamental groups of a special class of curves possessing outer singularities.
We consider \L ojasiewicz inequalities for a non-degenerate holomorphic function with an isolated... more We consider \L ojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the \L ojasiewicz 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 \L ojasiewicz non-degeneracy). We also introduce \L ojasiewicz inequality for strongly non-degenerate mixed functions and generalize this estimation for mixed functions.
The second author classified configurations of the singularities on tame sextics of torus type. I... more 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.
We give a criterion to test geometric properties such as Whitney equisingularity and Thom's a... more 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 two mixed curve C,C'ā C^2 which are defined by mixed functions of two variables z... more We consider two mixed curve C,C'ā C^2 which are defined by mixed functions of two variables z=(z_1,z_2). We have shown in 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 polynomials.
Mixed functions are analytic functions in variables z_1,..., z_n and their conjugates zĢ _1,..., z... more Mixed functions are analytic functions in variables z_1,..., z_n and their conjugates zĢ _1,..., zĢ _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.
We consider a mixed function of type H(z,zĢ )=f(z)gĢ (z) where f and g are convenient holomorphic f... more We consider a mixed function of type H(z,zĢ )=f(z)gĢ (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.
Let f(,) be a mixed strongly polar homogeneous polynomial of 3 variables =(z_1,z_2, z_3). It defi... more Let f(,) be a mixed strongly polar homogeneous polynomial of 3 variables =(z_1,z_2, z_3). It defines a Riemann surface V:={[]ā^2 | f(,)=0 } in the complex projective space ^2. We will show that for an arbitrary given g> 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 call polar weighted homogeneous polynomials of twisted join type.
where z = z1 1 Ā· Ā· Ā· z Ī½n n for Ī½ = (Ī½1, . . . , Ī½n) (respectively zĢ Ī¼ = zĢ1 1 Ā· Ā· Ā· zĢ Ī¼n n for... more where z = z1 1 Ā· Ā· Ā· z Ī½n n for Ī½ = (Ī½1, . . . , Ī½n) (respectively zĢ Ī¼ = zĢ1 1 Ā· Ā· Ā· zĢ Ī¼n n for Ī¼ = (Ī¼1, . . . , Ī¼n)) as usual. Here zĢj is the complex conjugate of zj. We call f(z, zĢ) a mixed analytic function (or a mixed polynomial, if f(z, zĢ) is a polynomial) of z1, . . . , zn. We are interested in the topology of the hypersurface V = {z ā C | f(z, zĢ) = 0}, which we call a mixed hypersurface. This approach is equivalent to the original one.
J(f, g) := fx(x, y)gy(x, y) ā fy(x, y)gx(x, y) where fx, fy are partial derivatives with respect ... more J(f, g) := fx(x, y)gy(x, y) ā fy(x, y)gx(x, y) where fx, fy are partial derivatives with respect to x and y respectively. We can also understand J(f, g) by (df ā§ dg)(x, y) = J(f, g) dx ā§ dy. Suppose that the mapping Ī¦ : C ā C is an automorphism in the sense that it has an inverse polynomial mapping ĪØ : C ā C. Then by the composition rule of the Jacobian, we have the equality J(Ī¦) ā” c for a non-zero constant c. The Jacobian conjecture is concerned with the converse assertion: (JC): If J(f, g) ā” c, c ā Cā, (f, g) : C ā C is an automorphism. The first essential contribution to this conjecture was made by Suzuki [4], Abhyankar [1] and the author did a small work about outside faces of mixed weights using Newton diagram [3]. There are many papers since then about this topic but it seems that there are few further progress on this conjecture of two variable case. This note is a rewritten version of [3] for a conference talk in Hanoi, October 2006. We tried to make some argument simpler th...
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