Topological invariants of higher order for a pair of plane curve germs

Abstract

Let ( g, f) be an analytic map germ from ( C 2,0) into ( C 2,0) and denote by ( u, v) the canonical coordinates in (g,f)( C 2) ; it is ( g( x, y), f( x, y))=( u, v). In [J. London Math. Soc. (2) 59 (1999) 207–226], we showed that the set constituted of the first (not necessarily characteristic) Puiseux exponent (in the ( u, v)-coordinates) of each branch δ of the discriminant curve of ( g, f) is a topological invariant of ( g, f). Here we prove that for each branch δ there exists an integer k( δ) such that the set constituted of the first (not necessarily characteristic) k( δ) exponents of the Puiseux series in the ( u, v)-coordinates of each δ is a topological invariant of ( g, f). We give different ways to compute these invariants.