On GCD(\Phi_N(a^n),\Phi_N(b^n))

TL;DR

This paper investigates the growth of the gcd of cyclotomic polynomial values at powers of multiplicatively independent elements, extending known bounds and exploring their behavior over different algebraic structures.

Contribution

It generalizes the lower bound for gcd of cyclotomic polynomial values for arbitrary N under GRH and extends results over finite fields without GRH.

Findings

Established a generalized lower bound for gcd(_N(a^n), _N(b^n)) under GRH.

Proved a corresponding lower bound over finite fields without GRH.

Extended previous bounds to arbitrary N for cyclotomic polynomial gcds.

Abstract

There has been interest during the last decade in properties of the sequence {gcd(a^n-1,b^n-1)}, n=1,2,3,..., where a,b are fixed (multiplicatively independent) elements in either the rational integers, the polynomials in one variable over the complex numbers, or the polynomials in one variable over a finite field. In the case of the rational integers, Bugeaud, Corvaja and Zannier have obtained an upper bound exp(\epsilon n) for any given \epsilon >0 and all large n, and demonstrate its approximate sharpness by extracting from a paper of Adleman, Pomerance, and Rumely a lower bound \exp(\exp(c\frac{log n}{loglog n})) for infinitely many n, where c is an absolute constant. The upper bound generalizes immediately to gcd(\Phi_N(a^n), \Phi_N(b^n)) for any positive integer N, where \Phi_N(x)$ is the Nth cyclotomic polynomial, the preceding being the case N=1. The lower bound has been generalized in the first author's Ph.D. thesis to N=2. In this paper we generalize the lower bound for arbitrary N but under GRH (the generalized Riemann Hypothesis). The analogue of the lower bound result for gcd(a^n-1,b^n-1) over F_q[T] was proved by Silverman; we prove a corresponding generalization (without GRH).