Abstract
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
, 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.
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hat is, F is continuous and F maps bounded sets to relatively compact sets.
A compact mapping sends bounded sets to relatively compact sets.
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© 2003 Springer Science+Business Media New York
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Krantz, S.G., Parks, H.R. (2003). Variations and Generalizations. In: The Implicit Function Theorem. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0059-8_5
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DOI: https://doi.org/10.1007/978-1-4612-0059-8_5
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6593-1
Online ISBN: 978-1-4612-0059-8
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