The Minimum Degree of Nonnegative Multiples of Cyclotomic Polynomials

Abstract

Let Phi_n(x) be the n-th cyclotomic polynomial, let p be the smallest prime divisor of n>1, and put P=n/p. We prove that every nonzero polynomial with nonnegative real coefficients divisible by Phi_n has degree at least (p-1)P, with equality exactly for the positive scalar multiples of 1+x^P+...+x^((p-1)P). This gives a uniform affirmative solution of Conjecture 1 in John P. Steinberger's 2012 paper for every integer n>1. The proof uses an explicit low-frequency trigonometric separation certificate and a unit-group rigidity argument. Status: Public Beta; internally verified candidate proof; external mathematical review pending.