manuscripta math. 39, 207- 241 (1985) manuscripta mathemati ca 9 Springer-Verlag 1985 SEMIGROUP OF TWO IRREDUCIBLE ALGEBROID PLANE CURVES by Valmecir Bayer (Ph.D. dissertation thesis presented at Instituto de Matem~tica Pura e Aplicada - IMPA - CNPq - Rio de Janeiro - Brasil) Introduction Let fl,f2,...,fn be irreducible series in k[[X,Y]], where k is an algebraically closed field. The semigroup S(fl,| associated to the irreducible algebroid plane curves (fl),...,(fn) is defined as the set of intersection vectors (I(fl~h), .... I(fn,h)) where (h) runs over al[ algebroid plane curves which do not have (fl) ~.o.~(fn) as branches. Waldi [8] showed that two singular points of algebraic plane curves are equisingular in the sense of Zariski [93 if and only if the semigroups associated to their respective branches are the same (up %o a permutation of the branches). Thus the problematic is in characterizing the semigroups associated to irreducible algebroid plane curves. In the case of a single branch, Angerm~ller [i] obtained one such characterization. In the case of two branches, Waldi showed that the semi- group S(fl~f2) is determined by the semigroups S(fl), S(f 2 ) 207