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

Let G = (4y2 + 2z)x2 + (10y2 + 6z) be the greatest common divisor (gcd) of two polynomials A, B ∈   [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.

• Type: Conference
• Year: 2005
• Where: ISSAC
• Authors: Jennifer de Kleine, Michael B. Monagan, Allan D. Wittkopf
• Keywords: ISSAC 2005, Modular Gcd Algorithms, Sparse Modular Gcd, Univariate Images