On Lehmer's Conjecture for Polynomials and for Elliptic Curves

TL;DR

This paper extends Lehmer's conjecture to certain polynomials over finite rings and elliptic curves, providing explicit bounds and new theoretical insights into their properties.

Contribution

It introduces a novel result for polynomials divisible by a specific form in (Z/mZ)[X] and formulates an analogous statement for elliptic curves.

Findings

Proves explicit bounds for polynomials divisible by 1+X+...+X^n in (Z/mZ)[X]

Establishes an analogous result for elliptic curves

Provides theoretical framework connecting polynomials and elliptic curves

Abstract

A number of authors have proven explicit versions of Lehmer's conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (Z/mZ)[X] by a polynomial of the form 1+X+...+X^n for some n > \epsilon*deg(f). We also formulate and prove an analogous statement for elliptic curves.