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Complex analysis again

  • Thread starter Mystic998
  • Start date
206
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1. Homework Statement

Let [itex]p(z)[/itex] be a polynomial of degree [itex]n \geq 1[/itex]. Show that [itex]\left\{z \in \mathbb{C} : \left|p(z)\right| > 1 \right\}[/tex] is connected with connectivity at most [itex]n+1[/itex].

2. Homework Equations

A region (connected, open set) considered as a set in the complex plane has finite connectivity n if its complement has n connected components in the extended complex plane.

3. The Attempt at a Solution

I'm not sure where to start, frankly. Showing the set is connected seems really tricky, though I'm admittedly probably overlooking something really obvious. As for connectivity, I think it has to do with the fact that the complement of the set is [itex]\left\{z \in \mathbb{C} : \left|p(z)\right| \leq 1\right\} \cup \left\{\infty\right\}[/itex] in the extended complex plane. So I think that because the polynomial has at most n roots, any preimage of the first set can have at most n disjoint connected sets mapped to it, then the point at infinity gives you one more connected component. But I'm not sure how to say that rigorously.
 

Answers and Replies

206
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Bump before bed
 

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