We consider the space Curv of complex affine lines t ↦ (x,y) = (ϕ(t),ψ(t)) with monic polynomials ϕ, ψ of fixed degrees and a map Expan from Curv to a complex affine space Puis with dim Curv = dim Puis, which is defined by initial Puiseux coefficients of the Puiseux expansion of the curve at infinity. We present some unexpected relations between geometrical properties of the curves (ϕ,ψ) and singularities of the map Expan. For example, the curve (ϕ,ψ) has a cuspidal singularity iff it is a critical point of Expan. We calculate the geometric degree of Expan in the cases gcd(degϕ,degψ) ≤ 2 and describe the non-properness set of Expan.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap93-3-7,
author = {Maciej Borodzik and Henryk \.Zo\l \k adek},
title = {Geometry of Puiseux expansions},
journal = {Annales Polonici Mathematici},
volume = {93},
year = {2008},
pages = {263-280},
zbl = {1136.14304},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap93-3-7}
}
Maciej Borodzik; Henryk Żołądek. Geometry of Puiseux expansions. Annales Polonici Mathematici, Tome 93 (2008) pp. 263-280. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap93-3-7/