Wydawnictwa / Banach Center Publications / Wszystkie tomy

Streszczenie

Let $\mathcal {O}_{f}$ be the local ring of an algebroid reduced curve $\{f=0\}$ over an algebraically closed field $K$, $\overline {\mathcal {O}}_{f} $ its integral closure in the total quotient ring of $\mathcal {O}_{f}$ and $\mathcal {C}_{f}$ the conductor of $\mathcal {O}_{f} $ in $\overline {\mathcal {O}}_{f} $. The codimension $c(f)=\dim_{K}\overline {\mathcal {O}}_{f} / \mathcal C_{f}$ is called the degree of singularity of the curve $\{f=0\}$. Suppose that the Newton polygon ${\mathcal N}_{f}$ of the curve $\{f=0\}$ intersects the axes at the points $(m,0), (0,n)$ and put $c({\mathcal N}_{f})=2$ (area of the polygon bounded by $\mathcal {N}_{f}$ and the axes$)+ ($number of integer points on $\mathcal {N}_{f})-m-n-1$. We prove that there exists a factorization $f=f_{1}\cdots f_{s}$ of $f$ in $K[[x,y]]$ such that $c(f)=c({\mathcal N}_{f})+\sum_{i=1}^{s}c(\tilde f_{i})$, where $\{\tilde f_{i}=0\}$ is obtained as a composition of quadratic transforms of the curve $\{f_{i}=0\}$. The proof is effective: the Newton polygon ${\mathcal N}_{f}$ and the initial parts of $f$ corresponding to the compact edges of ${\mathcal N}_{f}$ determine the Newton polygons of $f_{i}$ and the number of quadratic transforms necessary to compute $\tilde f_{i}$. As application of our result we give a formula for the Milnor number of $f$.