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

Authors:
Jennifer de Kleine;Michael Monagan;Allan Wittkopf
Affiliations:
Simon Fraser University, Burnaby, B.C. Canada;Simon Fraser University, Burnaby, B.C. Canada;Maplesoft, Waterloo, ON, Canada
Venue:
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Year:
2005

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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.