Number of Real Critical Points of Cyclotomic Polynomials

TL;DR

This paper investigates the number of real critical points of cyclotomic polynomials, establishing that for sufficiently separated primes, the derivative has exactly 2^k - 1 simple real roots.

Contribution

It provides a precise count of real critical points of cyclotomic polynomials under a separation condition on primes, a new insight into their critical point structure.

Findings

Number of real critical points is 2^k - 1 for sufficiently separated primes.

All real critical points are simple roots.

The result applies to cyclotomic polynomials with prime factors that are well-separated.

Abstract

We study the number of real critical points of a cyclotomic polynomial $\Phi_{n}(x)$, that is, the real roots of $\Phi_{n}^{\prime}(x)$. As usual, one can, without losing generality, restrict $n$ to be the product of distinct odd primes, say $p_{1}<\cdots<p_{k}$. We show that if the primes are "sufficiently separated" then there are exactly $2^{k}-1$ real roots of $\Phi_{n}^{\prime}(x)$ and each of them is simple.