Factorization of the polar curve and the Newton polygon. (English) Zbl 1083.32026
Consider \(f\in \mathbb C\{X,Y\}\), which has an isolated singularity. Consider a line \(bX-aY=0\) which is not tangent to \(f=0\), and consider the generic polar \( \partial f:=a \frac{\partial f}{\partial x} + b \frac{\partial f}{\partial y}\).
In this paper the authors study the factorization of \(\partial f\) using the Newton polygon of \(f\). As an application they calculate the minimal polar invariant and prove a bound on the number of special values in the pencil \(f -t\ell^N\).
In this paper the authors study the factorization of \(\partial f\) using the Newton polygon of \(f\). As an application they calculate the minimal polar invariant and prove a bound on the number of special values in the pencil \(f -t\ell^N\).
Reviewer: Theo de Jong (Mainz)
MSC:
32S55 | Milnor fibration; relations with knot theory |
14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |