Algorithms for the non-monic case of the sparse modular GCD algorithm

Abstract

Let G = (4y2+2z)x2 + (10y2+6z) be the greatest common divisor (Gcd) of two polynomials A, B ∈ ℤ[x,y,z]. Because G is not monic in the main variable x, the sparse modular Gcd algorithm of Richard Zippel cannot be applied directly as one is unable to scale univariate images of G in x consistently. We call this the normalization problem.We present two new sparse modular Gcd algorithms which solve this problem without requiring any factorizations. The first, a modification of Zippel's algorithm, treats the scaling factors as unknowns to be solved for. This leads to a structured coupled linear system for which an efficient solution is still possible. The second algorithm reconstructs the monic Gcd x2 + (5y2+3z)/(2y2+z) from monic univariate images using a sparse, variable at a time, rational function interpolation algorithm.