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## Complex #34: sneaky use of Rouche’s theorem

**Problem #34 is: Prove that there does not exist a polynomial of the form such that for all such that .**

**Proof:** Suppose that there does exist such a polynomial. Note that for . By Rouche’s theorem, has the same number of zeros (with multiplicities) as inside the unit disk. But that is impossible, since has zeros and has at most zeros. QED.

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Categories: Complex questions

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