Some results on counting roots of polynomials and the Sylvester resultant

TL;DR

This paper investigates the distribution of polynomial roots over modular integers and the Sylvester resultant, with applications to polynomial GCD computation and Diophantine equations.

Contribution

It introduces two new results on root distributions, enhancing understanding of polynomial behavior and computational methods in algebra.

Findings

Distribution of roots over integers modulo n analyzed

Distribution of roots of Sylvester resultants studied

Applications to polynomial GCD and Diophantine equations demonstrated

Abstract

We present two results, the first on the distribution of the roots of a polynomial over the ring of integers modulo $n$ and the second on the distribution of the roots of the Sylvester resultant of two multivariate polynomials. The second result has application to polynomial GCD computation and solving polynomial diophantine equations.