Polynomials with divisors of every degree

TL;DR

This paper investigates polynomials of the form t^n-1 to identify when they have divisors of every degree, establishing asymptotic bounds on the count of such integers up to x.

Contribution

It characterizes when polynomials t^n-1 have divisors of all degrees and proves asymptotic bounds for the number of such integers up to x.

Findings

Existence of constants c_1 and c_2 such that c_1 x/(log x) 8; F(x) 8; c_2 x/(log x)

Asymptotic density of integers n for which t^n-1 has divisors of every degree

Quantitative bounds on the count of such integers up to x

Abstract

We consider polynomials of the form t^n-1 and determine when members of this family have a divisor of every degree in Z[t]. With F(x) defined to be the number of such integers up to x, we prove the existence of two positive constants c_1 and c_2 such that $$c_1 x/(log x) \leq F(x) \leq c_2 x/(log x).$$