Journal of the Mathematical Society of Japan

Bifurcation sets of real polynomial functions of two variables and Newton polygons

Masaharu ISHIKAWA, Tat-Thang NGUYEN, and Tien-Son PHẠM

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Abstract

In this paper, we determine the bifurcation set of a real polynomial function of two variables for non-degenerate case in the sense of Newton polygons by using a toric compactification. We also count the number of singular phenomena at infinity, called “cleaving” and “vanishing”, in the same setting. Finally, we give an upper bound of the number of atypical values at infinity in terms of its Newton polygon. To obtain the upper bound, we apply toric modifications to the singularities at infinity successively.

Note

This work was supported by the Grant-in-Aid for Scientific Research (C), JSPS KAKENHI Grant Number 16K05140 and for Scientific Research (S), JSPS KAKENHI Grant Number 17H06128 and the National Foundation for Science and Technology Development (NAFOSTED), Grant number 101.04-2017.12 and 101.04-2016.05, Vietnam.

Article information

Source
J. Math. Soc. Japan, Volume 71, Number 4 (2019), 1201-1222.

Dates
Received: 8 May 2018
Revised: 21 June 2018
First available in Project Euclid: 13 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1560412821

Digital Object Identifier
doi:10.2969/jmsj/80518051

Mathematical Reviews number (MathSciNet)
MR4023304

Subjects
Primary: 32S20: Global theory of singularities; cohomological properties [See also 14E15]
Secondary: 32S15: Equisingularity (topological and analytic) [See also 14E15] 32S30: Deformations of singularities; vanishing cycles [See also 14B07]

Keywords
atypical value bifurcation set toric compactification

Citation

ISHIKAWA, Masaharu; NGUYEN, Tat-Thang; PHẠM, Tien-Son. Bifurcation sets of real polynomial functions of two variables and Newton polygons. J. Math. Soc. Japan 71 (2019), no. 4, 1201--1222. doi:10.2969/jmsj/80518051. https://projecteuclid.org/euclid.jmsj/1560412821


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