Computing Rational Zeros of Integral Polynomials by p-Adic Expansion

Abstract

We design, analyze and measure several algorithms for finding the rational zeros of univariate polynomials with integral coefficients of any size. The quadratic Newton-Hensel type algorithm behaves best in theory and practice. It has a maximum computing time of $O(n^3 L(d)^2 )$, where n is the degree and d measures the coefficient size of the given squarefree polynomial using classical arithmetic. We extend the method to Gaussian rational numbers.