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The question is, how many times does $f(C_r)$ go around $0$ for $C_r = \{x \in \mathbb{C}: |x|=r\}$ and a polynomial with complex coefficients $f$.

The answer is clear to me when $f(x) = ax^n + b$ but I don't see how exactly middle terms affect the result. How to see this without a graph? Can this be found for any $r$?

Many thanks

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Hint: It depends on the zeros of $f$. –  Alexander Thumm Aug 13 '12 at 17:00
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Should you decide to investigate such images deeper, see this paper and another one by the same author‌​. –  user31373 Aug 13 '12 at 17:51
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1 Answer 1

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Look at the Argument Principle (eg, Conway's, "Functions of one complex variable", V.3.4).

You wish to compute $n(f \circ \gamma, 0)$, where $\gamma(t) = r e^{i t}$, and $n$ is the winding number. The Argument Principle gives: $$n(f \circ \gamma, 0) = \sum n(\gamma, z_i) - \sum n(\gamma, p_j),$$ where $z_i$ are the zeros of $f$ and $p_j$ are the poles. Since $f$ is a polynomial, there are no poles, so you have $n(f \circ \gamma, 0) = \sum n(\gamma, z_i)$, ie, the number of zeros of $f$ inside $C_r$, counted according to multiplicity.

(The above assumes that $f$ has no zeros on the circle $C_r$.)

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