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Complex analysis again

  1. Mar 24, 2008 #1
    1. The problem statement, all variables and given/known data

    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. Relevant 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.
  2. jcsd
  3. Mar 24, 2008 #2
    Bump before bed
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