From Oil Fields to Hilbert Schemes

Kreuzer, Martin; Poulisse, H.N.J.; Robbiano, Lorenzo

doi:10.1007/978-3-211-99314-9_1

Cited by 7 publications

(4 citation statements)

References 17 publications

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“…Since border bases have been shown to provide good numerical stability (e.g., see [18] and [10]), they have been explored to study zero-dimensional systems with approximate coefficients obtained from empirical measurements. Several algorithms have been designed for computing border bases, for instance the algorithm presented in [8] and implemented in the ApCoCoA computer algebra system (cf.…”

Section: Introductionmentioning

confidence: 99%

Computing all border bases for ideals of points

2019

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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-Möller Algorithm [14], 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 [5] 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.Date: June 21, 2017.

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Section: Introductionmentioning

confidence: 99%

Computing all border bases for ideals of points

2019

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“…Border bases of zero-dimensional ideals have turned out to be a good alternative for Gröbner bases as they have nice numerical properties, see for instance [Ste04], [KPR09], [Mou07] and [MT08]. In the paper [KP11], border bases have been generalized to subideal border bases.…”

Section: Introductionmentioning

confidence: 99%

Module Border Bases

Kriegl

2013

Preprint

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In this paper, we generalize the notion of border bases of zerodimensional polynomial ideals to the module setting. To this end, we introduce order modules as a generalization of order ideals and module border bases of submodules with finite codimension in a free module as a generalization of border bases of zero-dimensional ideals in the first part of this paper. In particular, we extend the division algorithm for border bases to the module setting, show the existence and uniqueness of module border bases, and characterize module border bases analogously like border bases via the special generation property, border form modules, rewrite rules, commuting matrices, and liftings of border syzygies. Furthermore, we deduce Buchberger's Criterion for Module Border Bases and give an algorithm for the computation of module border bases that uses linear algebra techniques. In the second part, we further generalize the notion of module border bases to quotient modules. We then show the connection between quotient module border bases and special module border bases and deduce characterizations similar to the ones for module border bases. Moreover, we give an algorithm for the computation of quotient module border bases using linear algebra techniques, again. At last, we prove that subideal border bases are isomorphic to special quotient module border bases. This isomorphy immediately yields characterizations and an algorithm for the computation of subideal border bases.

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“…, g ν } such that (1) the unitary polynomials g i / g i vanish ε -approximately at X, where ε > 0 is a given threshold number, and (2) the normal remainders of the S-polynomials S(g i , g j ) for g i , g j with neighboring border terms are smaller than ε . Abstractly speaking, the last condition means that the point in the moduli space corresponding to G is "close" to the border basis scheme (see [11] and [8]). In practical applications, the AVI-algorithm turns out to be very stable and useful.…”

Section: Introductionmentioning

confidence: 99%

“…Unless explicitly stated otherwise, we use the notation and definitions of [9] and [10]. We shall assume that the reader has some familiarity with the theory of exact and approximate border bases (see for instance [5], [6], [7], [8], Section 6.4 of [10], and [12]).…”

Section: Introductionmentioning

confidence: 99%

Subideal border bases

Kreuzer

Poulisse

2011

Math. Comp.

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Abstract. In modeling physical systems, it is sometimes useful to construct border bases of 0-dimensional polynomial ideals which are contained in the ideal generated by a given set of polynomials. We define and construct such subideal border bases, provide some basic properties and generalize a suitable variant of the Buchberger-Möller algorithm as well as the AVI-algorithm of Heldt, Kreuzer, Pokutta, and Poulisse to the subideal setting. The subideal version of the AVI-algorithm is then applied to an actual industrial problem.

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From Oil Fields to Hilbert Schemes

Cited by 7 publications

References 17 publications

Computing all border bases for ideals of points

Computing all border bases for ideals of points

Module Border Bases

Subideal border bases

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