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Milnor number of a plane curve and Newton polygons. (English) Zbl 0997.32021
Summary: Given a local curve \(f=0\) we consider the Milnor number \(\mu(f)\) and the number of its irreducible components \(r(f)\). Suppose that \(f=0\) does not contain the axes.
It is well-known that for almost all \(f\) with the given Newton polygon \({\mathcal N}_f\) we have \(\mu(f)= \mu({\mathcal N}_f)\) and \(r(f)= r({\mathcal N}_f)\) where \(\mu({\mathcal N}_f)\) and \(r({\mathcal N}_f)\) depend only on \({\mathcal N}_f\). The aim of this note is to give an elementary proof of the inequality \(\mu(f)- \mu({\mathcal N}_f)\geq r({\mathcal N}_f)- r(f)\geq 0\) for all \(f\in \mathbb{C} [[X,Y]]\).

32S05 Local complex singularities
14B05 Singularities in algebraic geometry
14H20 Singularities of curves, local rings