Number of Real Critical Points of Cyclotomic Polynomials

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.

References in corpus (2)

• Flat Cyclotomic Polynomials: A New Approach
• Cyclotomic polynomials with prescribed height and prime number theory