Abstract
In deformations of polynomial functions one may encounter “singularity exchange at infinity” when singular points disappear from the space and produce “virtual” singularities which have an influence on the topology of the limit polynomial. We find several rules of this exchange phenomenon, in which the total quantity of singularity turns out to be not conserved in general.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
N. A’Campo, Le nombre de Lefschetz d’une monodromie, Indag. Math. 35 (1973), 113–118.
A. Andreotti, T. Frankel, The Lefschetz theorem on hyperplane sections, Ann. of Math. (2) 69 (1959), 713–717.
V.I. Arnol’d, Singularities of fractions and the behavior of polynomials at infinity, Tr. Mat. Inst. Steklova 221 (1998), 48–68; translation in Proc. Steklov Inst. Math. 1998, no. 2 (221), 40–59.
A. Bodin, Invariance of Milnor numbers and topology of complex polynomials, Comment. Math. Helv. 78 (2003), no. 1, 134–152.
A. Bodin, M. Tibăr, Topological triviality of families of complex polynomials, Adv. Math. 199 (2006), no. 1, 136–150.
S.A. Broughton, On the topology of polynomial hypersurfaces, Proceedings A.M.S. Symp. in Pure. Math., vol. 40, I (1983), 165–178.
S.M. Gusein-Zade, D. Siersma, Deformations of polynomials and their zeta functions, math.AG/0503450.
Hà H.V., A. Zaharia, Families of polynomials with total Milnor number constant, Math. Ann. 304 (1996), no. 3, 481–488.
F. Lazzeri, A theorem on the monodromy of isolated singularities, in: Singularités à Cargèse, 1972, pp. 269–275. Astérisque, Nos. 7 et 8, Soc. Math. France, Paris, 1973.
Lê D.T., Une application d’un théorème d’A’Campo à l’équisingularité, Nederl. Akad. Wetensch. Proc. (Indag. Math.) 35 (1973), 403–409.
A. Némethi, C. Sabbah, Semicontinuity of the spectrum at infinity, Abh. Math. Sem. Univ. Hamburg 69 (1999), 25–35.
A. Parusiński, On the bifurcation set of complex polynomial with isolated singularities at infinity, Compositio Math. 97 (1995), no. 3, 369–384.
D. Siersma, J. Smeltink, Classification of singularities at infinity of polynomials of degree 4 in two variabales, Georgian Math. J. 7(1) (2000), 179–190.
D. Siersma, M. Tibăr, Singularities at infinity and their vanishing cycles, Duke Math. Journal 80(3) (1995), 771–783.
D. Siersma, M. Tibăr, Deformations of polynomials, boundary singularities and monodromy, Mosc. Math. J. 3(2) (2003), 661–679.
M. Tibăr, Bouquet decomposition of the Milnor fibre, Topology 35,1 (1996), 227–241.
M. Tibăr, On the monodromy fibration of polynomial functions with singularities at infinity, C.R. Acad. Sci. Paris, 324 (1997), 1031–1035.
M. Tibăr, Regularity at infinity of real and complex polynomial maps, in: Singularity Theory, The C.T.C Wall Anniversary Volume, LMS Lecture Notes Series 263 (1999), 249–264. Cambridge University Press.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Siersma, D., Tibăr, M. (2006). Singularity Exchange at the Frontier of the Space. In: Brasselet, JP., Ruas, M.A.S. (eds) Real and Complex Singularities. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7776-2_23
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7776-2_23
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7775-5
Online ISBN: 978-3-7643-7776-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)