Variations and Generalizations

In Section 2.2, we described the method Newton devised for examining the local behavior of the locus of points satisfying a polynomial equation in two variables. It is easy to extend that method to the locus of an equation of the form 5.1$$ {z^m} + {a_{{m - 1}}}(w){z^{{m - 1}}} + {a_{{m - 2}}}(w){z^{{m - 2}}} + \cdots + {a_0}(w) = 0 $$, where each a _i (w) is a holomorphic function of $$ w \in \mathbb{C} $$ that vanishes at $$ w = 0 $$. Such an extension is significant, because the Weierstrass preparation theorem will show us that the behavior near (0, 0) of the locus of an equation of the form given in (5.1) is completely representative of the local behavior of the locus of points satisfying $$ F\left( {w,z} \right) = 0 $$, where F is holomorphic.