Article

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

Authors:

Jennifer de Kleine,

Michael Monagan,

Allan WittkopfAuthors Info & Claims

ISSAC '05: Proceedings of the 2005 international symposium on Symbolic and algebraic computation

Pages 124 - 131

https://doi.org/10.1145/1073884.1073903

Published: 24 July 2005 Publication History

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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.

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Cited By

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Monagan MHuang Q(2024)A New Sparse Polynomial GCD by Separating TermsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669684(134-142)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669684
Asadi MBrandt AMoir RMaza MXie Y(2022)Parallelization of Triangular Decompositions: Techniques and ImplementationJournal of Symbolic Computation10.1016/j.jsc.2022.08.015Online publication date: Aug-2022
https://doi.org/10.1016/j.jsc.2022.08.015
van der Hoeven JLecerf G(2021)On sparse interpolation of rational functions and gcdsACM Communications in Computer Algebra10.1145/3466895.346689655:1(1-12)Online publication date: 20-May-2021
https://dl.acm.org/doi/10.1145/3466895.3466896
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Index Terms

Algorithms for the non-monic case of the sparse modular GCD algorithm
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on polynomials

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Published In

ISSAC '05: Proceedings of the 2005 international symposium on Symbolic and algebraic computation

July 2005

388 pages

ISBN:1595930957

DOI:10.1145/1073884

General Chairs:
Xiao-Shan Gao
MMRC, Chinese Academy of Sciences, China
,
George Labahn
University of Waterloo, ON, Canada

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 24 July 2005

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https://dl.acm.org/doi/10.1145/3666000.3669684
Asadi MBrandt AMoir RMaza MXie Y(2022)Parallelization of Triangular Decompositions: Techniques and ImplementationJournal of Symbolic Computation10.1016/j.jsc.2022.08.015Online publication date: Aug-2022
https://doi.org/10.1016/j.jsc.2022.08.015
van der Hoeven JLecerf G(2021)On sparse interpolation of rational functions and gcdsACM Communications in Computer Algebra10.1145/3466895.346689655:1(1-12)Online publication date: 20-May-2021
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A sparse modular GCD algorithm for polynomials over algebraic function fields

A Fast Parallel Sparse Polynomial GCD Algorithm

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