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Paper   IPM / M / 15383
School of Mathematics
  Title:   Computing all border bases for ideals of points
  Author(s):  Amir Hashemi (Joint with M. Kreuzer)
  Status:   Published
  Journal: J. Algebra Appl.
  Vol.:  18
  Year:  2019
  Pages:   # 23 Pages
  Supported by:  IPM
In this paper, we consider the problem of computing all possible order ideals and also sets connected to 1, and the corresponding border bases, for the vanishing ideal of a given finite set of points. In this context, two different approaches are discussed: based on the Buchberger�??Moller Algorithm [H. M. Moller and B. Buchberger, The construction of multivariate polynomials with preassigned zeros, EUROCAM �??82 Conf., Computer Algebra, Marseille/France 1982, Lect. Notes Comput. Sci. 144, (1982), pp. 24�??31], we first propose a new algorithm to compute all possible order ideals and the corresponding border bases for an ideal of points. The second approach involves adapting the Farr�??Gao Algorithm [J. B. Farr and S. Gao, Computing Grobner bases for vanishing ideals of finite sets of points, in 16th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC-16, Las Vegas, NV, USA (Springer, Berlin, 2006), pp. 118�??127] for finding all sets connected to 1, as well as the corresponding border bases, for an ideal of points. It should be noted that our algorithms are term ordering free. Therefore, they can compute successfully all border bases for an ideal of points. Both proposed algorithms have been implemented and their efficiency is discussed via a set of benchmarks.

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