manuscripta math. 37, I- 9 (1982) manuscripta mathematica 9 Springer-Verlag 1982 AN INTEGRAL CRITERION FOR THE EQUIVALENCE OF PLANE CURVES Alicia DICKENSTEIN and Carmen SESSA We give in this paper a necessary and sufficient condi- tion for the Zariski equivalence of algebroid plane ir- reducible curves C ~ (f=0) over an algebraically closed field of characteristic zero and, in the analytic case, an integral expression for the characteristic numbers. We express these numbers as the intersection multiplici - ties of (f=0) with a family of cycles associated to f, and also as residues of meromorphic differentials on C. Definitions and notation Let C E (f(x,y) = 0 ) , f s k [[ x, y ]] irreducible and suppose (x=0) is not a tangent. Let x=t n y = ~(t) = i a:t be a parametrization of C. We will assume C i>~ • slngular (n > i) and G.C.D. {n,i : a. # 0} = I. The 1 characteristic of C is the g+l - tuple (n;BI,B2,...,Bg), where n--mult C ; eo=n ; 8j+l=min {ieN: ~i, ai#0} and e.=G.C.D {n,Bl,...,Bj} (I < j < g) , e = i. 3 -- _ g ei=l Let us define G.= { T s k : r } 0 < i < g. It is clear that Go_ ~ G1 ~-''" ~-G i Gi+l... ~_ Gg= {i} and i G'I - Gi+l I = e.l - ei+ 1 , where IAI denotes the card- inal of the set A. LEMMA i: Let x=t n zation of C ; then ord t(~) - ~(rt)) Z a.t i y= ~(t) = i>n l be a parametri - = Bi+ 1 if r e G l Gi+ 1 . 0025-2611/82/0037/0001/$0t.80