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The Vanishing Ideal of a Finite Set of Points with Multiplicity Structures

In: Computer Mathematics

Author

Listed:
  • Na Lei

    (Jilin University, School of Mathematics)

  • Xiaopeng Zheng

    (Jilin University, School of Mathematics)

  • Yuxue Ren

    (Jilin University, School of Mathematics)

Abstract

Given a finite set of arbitrarily distributed points in affine space with multiplicity structures, we present an algorithm to compute the reduced Gr $${\ddot{\mathrm{o}}}$$ o ¨ bner basis of the vanishing ideal under the lexicographic order. We split the problem into several smaller ones which can be solved by induction over variables and then use our new algorithm for intersection of ideals to compute the result of the original problem. The new algorithm for intersection of ideals is mainly based on the Extended Euclidean Algorithm. Our method discloses the essential geometric connection between the relative position of the points with multiplicity structures and the leading monomials of the reduced Gr $${\ddot{\mathrm{o}}}$$ o ¨ bner basis of the vanishing ideal.

Suggested Citation

  • Na Lei & Xiaopeng Zheng & Yuxue Ren, 2014. "The Vanishing Ideal of a Finite Set of Points with Multiplicity Structures," Springer Books, in: Ruyong Feng & Wen-shin Lee & Yosuke Sato (ed.), Computer Mathematics, edition 127, pages 275-296, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-43799-5_21
    DOI: 10.1007/978-3-662-43799-5_21
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