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

Let (g,f) be an analytic map germ from (C2,0) into (C2,0) and denote by (u,v) the canonical coordinates in (g,f)(C2); 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.