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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 $$f\in \mathbb{C} [[X,Y]]$$.

##### MSC:
 32S05 Local complex singularities 14B05 Singularities in algebraic geometry 14H20 Singularities of curves, local rings
##### Keywords:
plane curve; Milnor number; Newton polygon