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.