Newton polyhedra and order of contact on real hypersurfaces

Jacobian Newton Polyhedra and equisingularity

This is the LaTeX version of the handwritten notes of a lecture at the Kyoto Singularities Symposium held at the RIMS in April 1978. It presents the relationship of various invariants of isolated singularities of complex analytic hypersurfaces with the jacobian Newton polygon built from the polar curves. An appendix presents features of the semiring of Newton polygons, because the product of Jacobian Newton polygons corresponds to the Thom-Sebastiani sum of hypersurfaces. Analogies between mixed multiplicities and mixed volumes are noticed. They have been explained since.

Lê's polyhedron for line singularities

We study the topology of a line singularity, which is a complex hypersurface with non-isolated singularity given by a complex line. We describe the degeneration of its Milnor fibre to the singular hypersurface by means of a pair of polyhedra, one in the Milnor fibre and other in the singular fibre, which are deformation retracts of the corresponding fibres; and a continuous map taking the Milnor fibre to the singular fibre and the first polyhedron to the second one, which restrict to a homeomorphism outside the polyhedra. In the same sense, we also study the topology of a complex isolated sing…

On holomorphic curves tangent to real hypersurfaces of infinite type

The purpose of this paper is to investigate the geometric properties of real hypersurfaces of D'Angelo infinite type in ${\mathbb C}^n$. In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using Newton polyhedra,which is an important concept in singularity theory. More precisely,equivalence conditions are given in the case of some model hypersurfaces.

Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude

The asymptotic behavior at infinity of oscillatory integrals is in detail investigated by using the Newton polyhedra of the phase and the amplitude. We are especially interested in the case that the amplitude has a zero at a critical point of the phase. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation.

Lower bound of Newton number

We show a lower estimate of the Milnor number of an isolated hypersurface singularity, via its Newton number. We also obtain analogous estimate of the Milnor number of an isolated singularity of a similar complete intersection variety.

位于 $\mathbb{C}^2$ 中某些 Levi 平坦超曲面上的紧集的多项式凸性

Polynomial convexity of compacts that lies in certain Levi-flat hypersurfaces in $\mathbb{C}^2$

本文证明了位于某些 Levi 平坦超曲面上的全实圆盘的多项式凸性，并给出了 Hartog 三角形边界上全实圆盘多项式凸性的充要条件及一般紧子集的充分条件。