## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Singularities in Geometry and Topology 2011, V. Blanlœil and O. Saeki, eds. (Tokyo: Mathematical Society of Japan, 2015), 173 - 202

### Mixed functions of strongly polar weighted homogeneous face type

#### Abstract

Let $f(\mathbf{z}, \bar{\mathbf{z}})$ be a mixed polynomial with strongly non-degenerate face functions. We consider a canonical toric modification $\pi: X\to\mathbb{C}^n$ and a polar modification $\pi_{\mathbb{R}}: Y\to X$. 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.

#### Article information

**Source***Singularities in Geometry and Topology 2011*, V. Blanlœil and O. Saeki, eds. (Tokyo: Mathematical Society of Japan, 2015), 173-202

**Dates**

Received: 26 January 2012

Revised: 19 July 2012

First available in Project Euclid:
19 October 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1539916287

**Digital Object Identifier**

doi:10.2969/aspm/06610173

**Mathematical Reviews number (MathSciNet)**

MR3382050

**Zentralblatt MATH identifier**

1360.32028

**Subjects**

Primary: 14P05: Real algebraic sets [See also 12D15, 13J30] 32S55: Milnor fibration; relations with knot theory [See also 57M25, 57Q45]

**Keywords**

Strongly polar weighted homogeneous Milnor fibration toric modification

#### Citation

Oka, Mutsuo. Mixed functions of strongly polar weighted homogeneous face type. Singularities in Geometry and Topology 2011, 173--202, Mathematical Society of Japan, Tokyo, Japan, 2015. doi:10.2969/aspm/06610173. https://projecteuclid.org/euclid.aspm/1539916287