On degrees of modular common divisors and the Big prime gcd algorithm

TL;DR

This paper introduces improved algorithms for computing polynomial gcds over integers by leveraging bounds on degrees of common divisors, estimates of prime divisors, and auxiliary primes, enhancing efficiency and robustness.

Contribution

The paper proposes novel modifications to the Big prime modular gcd algorithm, incorporating degree bounds and prime divisor estimates for better performance.

Findings

Algorithms successfully applied to polynomials with intermediate expression swell

Improved gcd computation efficiency demonstrated

Enhanced detection of coprime polynomials

Abstract

We consider a few modifications of the Big prime modular $\gcd$ algorithm for polynomials in $\Z[x]$. Our modifications are based on bounds of degrees of modular common divisors of polynomials, on estimates of the number of prime divisors of a resultant and on finding preliminary bounds on degrees of common divisors using auxiliary primes. These modifications are used to suggest improved algorithms for $\gcd$ calculation and for coprime polynomials detection. To illustrate the ideas we apply the constructed algorithms on certain polynomials, in particular, on polynomials from Knuth's example of intermediate expression swell.