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Computing Sparse GCD of Multivariate Polynomials via Polynomial Interpolation

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Abstract

The problem of computing the greatest common divisor (GCD) of multivariate polynomials, as one of the most important tasks of computer algebra and symbolic computation in more general scope, has been studied extensively since the beginning of the interdisciplinary of mathematics with computer science. For many real applications such as digital image restoration and enhancement, robust control theory of nonlinear systems, L1-norm convex optimization in compressed sensing techniques, as well as algebraic decoding of Reed-Solomon and BCH codes, the concept of sparse GCD plays a core role where only the greatest common divisors with much fewer terms than the original polynomials are of interest due to the nature of problems or data structures. This paper presents two methods via multivariate polynomial interpolation which are based on the variation of Zippel’s method and Ben-Or/Tiwari algorithm, respectively. To reduce computational complexity, probabilistic techniques and randomization are employed to deal with univariate GCD computation and univariate polynomial interpolation. The authors demonstrate the practical performance of our algorithms on a significant body of examples. The implemented experiment illustrates that our algorithms are efficient for a quite wide range of input.

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Acknowledgements

The authors are grateful to YANG Zhengfeng at East China Normal University. We would also like to thank anonymous reviewers for their very valuable comments.

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Authors and Affiliations

  1. School of Mathematics and Computing Science, Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, 541004, China

    Min Tang

  2. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, China

    Min Tang

  3. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Bingyu Li

  4. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Zhenbing Zeng

Authors
  1. Min Tang
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  2. Bingyu Li
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  3. Zhenbing Zeng
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Corresponding author

Correspondence to Min Tang.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant Nos. 11471209, 11561015, and 11301066, and Guangxi Key Laboratory of Cryptography and Information Security under Grant No. GCIS201615.

This paper was recommended for publication by Editor ZHANG Yang.

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Tang, M., Li, B. & Zeng, Z. Computing Sparse GCD of Multivariate Polynomials via Polynomial Interpolation. J Syst Sci Complex 31, 552–568 (2018). https://doi.org/10.1007/s11424-017-6332-0

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  • DOI: https://doi.org/10.1007/s11424-017-6332-0

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