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Simultaneously Computing a Maximal Independent Set Modulo an Ideal and a Gröbner Basis of the Ideal

Author

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  • Ping Liu

    (School of Mathematics, Key Laboratory of Symbolic Computation and Knowledge Engineering (Ministry of Education), Jilin University, Changchun 130012, China)

  • Baoxin Shang

    (College of Science, Northeast Electric Power University, Jilin 132012, China)

  • Shugong Zhang

    (School of Mathematics, Key Laboratory of Symbolic Computation and Knowledge Engineering (Ministry of Education), Jilin University, Changchun 130012, China)

Abstract

To solve problems on a positive-dimensional ideal, I ⊂ k [ X ] , a maximal independent set U ⊂ X modulo I , and a Gröbner basis of I e , where I e is the extension of I to k ( U ) [ V ] ( V : = X ∖ U ) , are widely used. As far as we know, they are usually computed separately, i.e., U is calculated first and the Gröbner basis is computed after U is obtained. In this paper, we present an efficient algorithm for computing a maximal independent set U modulo I , and a Gröbner basis of I e simultaneously. Differently from computing them separately, the algorithm takes full advantage of the polynomial information throughout the Gröbner basis computation to obtain U as soon as possible; hence, it significantly improves the computing efficiency.

Suggested Citation

  • Ping Liu & Baoxin Shang & Shugong Zhang, 2025. "Simultaneously Computing a Maximal Independent Set Modulo an Ideal and a Gröbner Basis of the Ideal," Mathematics, MDPI, vol. 13(18), pages 1-14, September.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:18:p:3037-:d:1753922
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