A DECOMPOSITION THEOREM FOR THE GENERIC P OLARS OF A PLANE CURVE

In this Note Lye state a decomposition theorem into bunches of branche s (i.e. analytically irreducible germs) for the generic polar curves o f a reduced germ of a plane analytic curve, with equation f(x,y) = 0. They are the curves with equation partial derivative f(x,y)/partial de rivative y + tau partial derivative f(x,y)/partial derivative x = 0 wi th generic tau. All the branches of the same bunch have tile same cont act with each branch of C. A number of the first terms of the Puiseux expansion of each branch of the polar is therefore independent of tau; this number depends only oil the bunch to which the branch belongs. T his generalizes results of H.J.S. Smith [8], M. Merle [7] (where C is a branch), E. Casas [1], F. Delgado [2]. We show by ail example that i t is also optimal. We have shown elsewhere that it implies the results of Le-Michel-Weber [6].