A note on the plane curve singularities in positive characteristic

Given an algebroid plane curve f = 0 over an algebraically closed field of characteristic p >= 0 we consider the Milnor number mu (f), the delta invariant delta(f) and the number r(f) of its irreducible components. Put (sic)(f) = 2 delta(f)- r(f) + 1. If p = 0 then (sic)(f ) = mu (f) (the Milnor formula). If p > 0 mu (f ) is not an invariant and mu (f) plays the role of mu (f). Let Nf be the Newton polygon of f. We define the numbers mu (N-f) and r(N-f) which can be computed by explicit formulas. The aim of this note is to give a simple proof of the inequality (sic)(f)- mu (N-f) >= r(N-f )- r(f) >= 0 due to Boubakri, Greuel and Markwig. We also prove that mu (f) = mu (N-f) when f is non-degenerate.