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October, 2019 Bifurcation sets of real polynomial functions of two variables and Newton polygons
Masaharu ISHIKAWA, Tat-Thang NGUYEN, Tien-Son PHẠM
J. Math. Soc. Japan 71(4): 1201-1222 (October, 2019). DOI: 10.2969/jmsj/80518051


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.

Funding Statement

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.


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Masaharu ISHIKAWA. Tat-Thang NGUYEN. Tien-Son PHẠM. "Bifurcation sets of real polynomial functions of two variables and Newton polygons." J. Math. Soc. Japan 71 (4) 1201 - 1222, October, 2019.


Received: 8 May 2018; Revised: 21 June 2018; Published: October, 2019
First available in Project Euclid: 13 June 2019

zbMATH: 07174403
MathSciNet: MR4023304
Digital Object Identifier: 10.2969/jmsj/80518051

Primary: 32S20
Secondary: 32S15 , 32S30

Keywords: atypical value , bifurcation set , toric compactification

Rights: Copyright © 2019 Mathematical Society of Japan


Vol.71 • No. 4 • October, 2019
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