On the Joint Distribution of the Roots of Pairs of Polynomial Congruences

TL;DR

This paper extends Hooley's 1964 result by demonstrating that pairs of roots from two primitive irreducible polynomials are jointly uniformly distributed in the unit torus, revealing their joint distribution properties.

Contribution

It proves the joint uniform distribution of roots of two primitive irreducible polynomials, generalizing previous results on single polynomial root distributions.

Findings

Pairs of roots are jointly uniformly distributed in the unit torus.

The distribution of roots of two polynomials behaves independently in the limit.

Generalization of Hooley's result to pairs of polynomial roots.

Abstract

Let f(x) be a primitive irreducible polynomial with integer coefficients of degree greater than one. In 1964, Hooley showed that the sequence of normalized roots u/n, where f(u) = 0(n), ordered in the obvious way, is uniformly distributed modulo one. It is the goal of this paper to show that if f(x) and g(x) are a pair of primitive irreducible polynomials of degree greater than one, not necessarily distinct, then the sequence (u/n,v/n), with f(u) = 0(n) and g(v) = 0(n), ordered in the obvious way, is uniformly distributed modulo one in the unit torus.