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An Improvement of the Rational Representation for High-Dimensional Systems

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Abstract

Based on the rational univariate representation of zero-dimensional polynomial systems, Tan and Zhang proposed the rational representation theory for solving a high-dimensional polynomial system, which uses so-called rational representation sets to describe all the zeros of a high-dimensional polynomial system. This paper is devoted to giving an improvement for the rational representation. The idea of this improvement comes from a minimal Dickson basis used for computing a comprehensive Gröbner system of a parametric polynomial system to reduce the number of branches. The authors replace the normal Grobner basis G satisfying certain conditions in the original algorithm (Tan-Zhang’s algorithm) with a minimal Dickson basis Gm of a Grobner basis for the ideal, where Gm is smaller in size than G. Based on this, the authors give an improved algorithm. Moreover, the proposed algorithm has been implemented on the computer algebra system Maple. Experimental data and its performance comparison with the original algorithm show that it generates fewer branches and the improvement is rewarding.

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Authors and Affiliations

  1. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Fanghui Xiao & Dingkang Wang

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Fanghui Xiao & Dingkang Wang

  3. Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, 100191, China

    Dong Lu

  4. School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    Dong Lu

  5. College of Science, China Agricultural University, Beijing, 100083, China

    Xiaodong Ma

Authors
  1. Fanghui Xiao
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  2. Dong Lu
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  3. Xiaodong Ma
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  4. Dingkang Wang
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Corresponding author

Correspondence to Dong Lu.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant No. 11801558, the Chinese Universities Scientific Funds under Grant No. 15059002 and the CAS Key Project QYZDJ-SSW-SYS022.

This paper was recommended for publication by Editor FENG Ruyong.

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Xiao, F., Lu, D., Ma, X. et al. An Improvement of the Rational Representation for High-Dimensional Systems. J Syst Sci Complex 34, 2410–2427 (2021). https://doi.org/10.1007/s11424-020-9316-4

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  • DOI: https://doi.org/10.1007/s11424-020-9316-4

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