@article{5f99ebafb7e448e2bbff2625c8ededdf,

title = "Bifurcation sets of real polynomial functions of two variables and Newton polygons",

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.",

keywords = "Atypical value, Bifurcation set, Toric compactification",

author = "Masaharu Ishikawa and Nguyen, {Tat Thang} and Pham, {Ti{\^e}n Son}",

note = "Funding Information: 2010 Mathematics Subject Classification. Primary 32S20; Secondary 32S15, 32S30. Key Words and Phrases. atypical value, bifurcation set, toric compactification. 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. Funding Information: 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. The second author is a member of the project ?Joint study in Analysis and Geometry?, ICRTM01 2019.05, conducted at the International Centre for Research and Training in Mathematics, Institute of Mathematics, VAST. He would like to thank other members for useful discussion. A part of this work was performed while the second author and the third author had been visiting at Vietnam Institute for Advanced Study in Mathematics (VIASM), Hanoi, Vietnam. These two authors would like to thank the Institute for financial support and excellent working conditions.",

year = "2019",

doi = "10.2969/jmsj/80518051",

language = "English",

volume = "71",

pages = "1201--1222",

journal = "Journal of the Mathematical Society of Japan",

issn = "0025-5645",

publisher = "Mathematical Society of Japan - Kobe University",

number = "4",

}