Trivial Estimate for Cyclotomic Polynomials
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Theorem
Let $n \ge 1$ be a natural number.
Let $\Phi_n$ be the $n$th cyclotomic polynomial.
Let $\phi$ be the Euler totient function.
Let $z \in \C$ be a complex number.
Then:
- $\size {\size z - 1}^{\map \phi n} \le \size {\map {\Phi_n} z} \le \paren {\size z + 1}^{\map \phi n}$
where:
- the first inequality becomes an equality only if:
- $n = 1$ and $z \in \R_{\ge 0}$
- or:
- $n = 2$ and $z \in \R_{\le 0}$
- the second inequality becomes an equality only if:
- $n = 1$ and $z \in \R_{\le 0}$
- or:
- $n = 2$ and $z \in \R_{\ge 0}$
Proof
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