### Abstract

A. Némethi and A. Zaharia have defined the explicit set for a complex polynomial function f: C^{n} → C and conjectured that the bifurcation set of the global fibration of f is given by the union of the set of critical values and the explicit set of f. They have proved only the case n = 2 and f is Newton non-degenerate. In the present paper we will prove this for the case n = 2, containing the Newton degenerate case, by using toric modifications and give an expression of the bifurcation set of f in the words of Newton polygons.

Original language | English |
---|---|

Pages (from-to) | 161-196 |

Number of pages | 36 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 54 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2002 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Bifurcation set
- Complex polynomial functions
- Newton polygons
- Singularities at infinity
- Toric modifications

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**The bifurcation set of a complex polynomial function of two variables and the newton polygons of singularities at infinity.** / Ishikawa, Masaharu.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The bifurcation set of a complex polynomial function of two variables and the newton polygons of singularities at infinity

AU - Ishikawa, Masaharu

PY - 2002/1/1

Y1 - 2002/1/1

N2 - A. Némethi and A. Zaharia have defined the explicit set for a complex polynomial function f: Cn → C and conjectured that the bifurcation set of the global fibration of f is given by the union of the set of critical values and the explicit set of f. They have proved only the case n = 2 and f is Newton non-degenerate. In the present paper we will prove this for the case n = 2, containing the Newton degenerate case, by using toric modifications and give an expression of the bifurcation set of f in the words of Newton polygons.

AB - A. Némethi and A. Zaharia have defined the explicit set for a complex polynomial function f: Cn → C and conjectured that the bifurcation set of the global fibration of f is given by the union of the set of critical values and the explicit set of f. They have proved only the case n = 2 and f is Newton non-degenerate. In the present paper we will prove this for the case n = 2, containing the Newton degenerate case, by using toric modifications and give an expression of the bifurcation set of f in the words of Newton polygons.

KW - Bifurcation set

KW - Complex polynomial functions

KW - Newton polygons

KW - Singularities at infinity

KW - Toric modifications

UR - http://www.scopus.com/inward/record.url?scp=0036068942&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036068942&partnerID=8YFLogxK

U2 - 10.2969/jmsj/1191593959

DO - 10.2969/jmsj/1191593959

M3 - Article

AN - SCOPUS:0036068942

VL - 54

SP - 161

EP - 196

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 1

ER -