Greatest common divisors of values of integer polynomials and an application to maximal tori

TL;DR

This paper presents a method to compute the gcd of polynomial values at integers and applies it to classify the structure of maximal tori in exceptional Lie groups.

Contribution

It introduces a technique for gcd computation of polynomial values and applies it to determine maximal tori structures in Lie groups.

Findings

Method for gcd of polynomial values at integers

Classification of maximal tori in exceptional Lie groups

Explicit elementary divisor structures

Abstract

We describe how to compute for two polynomials $f(X), g(X) \in \mathbb{Z}[X]$ with integer coefficients the greatest common divisors of $f(z)$ and $g(z)$ for all integers $z \in \mathbb{Z}$. As an application we determine the structures (elementary divisors) of all maximal tori in exceptional groups of Lie type.