Home > Complex questions > Complex #34: sneaky use of Rouche’s theorem

Complex #34: sneaky use of Rouche’s theorem

Problem #34 is: Prove that there does not exist a polynomial of the form p(z)= z^n+a_{n-1}z^{n-1}+\ldots+a_0 such that |p (z)| < 1 for all z such that |z|=1.

Proof: Suppose that there does exist such a polynomial. Note that |z^n|>|-p(z)| for |z|=1. By Rouche’s theorem, z^n-p(z)=-a_{n-1}z^{n-1}-\ldots-a_0 has the same number of zeros (with multiplicities) as z^n inside the unit disk. But that is impossible, since z^n has n zeros and z^n-p(z) has at most n-1 zeros. QED.

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Categories: Complex questions
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