Characterizations of border bases

Algorithms and Computation in Mathematics, 2005

This chapter is devoted to laying the algebraic foundations for border bases of ideals. Using an order ideal O, we describe a zero-dimensional ideal from the outside. The first and higher borders of O can be used to measure the distance of a term from O and to define O-border bases. We study their existence and uniqueness, their relation to Gröbner bases, and their characterization in terms of commuting matrices. Finally, we use border bases to solve a problem coming from statistics. Proposition 4.3.9. Let O be an order ideal such that the residue classes of the elements of O form a K-vector space basis of P/I. Let G be the O-border basis of I, and let G be the subset of G consisting of the elements marked by the corners of O. Then the following conditions are equivalent. 1. There exists a term ordering σ such that O = O σ (I). 2. The elements in G are marked coherently.

Journal of Algebra and Its Applications, 2019

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 [H. M. Möller 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 Gröbner 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 ...

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation - ISSAC '12, 2012

In this paper, we generalized the construction of border bases to non-zero dimensional ideals for normal forms compatible with the degree, tackling the remaining obstacle for a general application of border basis methods. First, we give conditions to have a border basis up to a given degree. Next, we describe a new stopping criteria to determine when the reduction with respect to the leading terms is a normal form. This test based on the persistence and regularity theorems of Gotzmann yields a new algorithm for computing a border basis of any ideal, which proceeds incrementally degree by degree until its regularity. We detail it, prove its correctness, present its implementation and report some experimentations which illustrate its practical good behavior.

Collectanea mathematica, 2009

Hilbert schemes of zero-dimensional ideals in a polynomial ring can be covered with suitable affine open subschemes whose construction is achieved using border bases. Moreover, border bases have proved to be an excellent tool for describing zero-dimensional ideals when the coefficients are inexact. And in this situation they show a clear advantage with respect to Gröbner bases which, nevertheless, can also be used in the study of Hilbert schemes, since they provide tools for constructing suitable stratifications. In this paper we compare Gröbner basis schemes with border basis schemes. It is shown that Gröbner basis schemes and their associated universal families can be viewed as weighted projective schemes. A first consequence of our approach is the proof that all the ideals which define a Gröbner basis scheme and are obtained using Buchberger's Algorithm, are equal. Another result is that if the origin (i.e. the point corresponding to the unique monomial ideal) in the Gröbner basis scheme is smooth, then the scheme itself is isomorphic to an affine space. This fact represents a remarkable difference between border basis and Gröbner basis schemes. Since it is natural to look for situations where a Gröbner basis scheme and the corresponding border basis scheme are equal, we address the issue, provide an answer, and exhibit some consequences. Open problems are discussed at the end of the paper. K such that P/I has a fixed basis. What is interesting is that the construction of such subschemes is performed using border bases (see for instance [7], [8], and [14]). Source 3. Despite their inability to treat inexact data well, Gröbner bases can nevertheless be used in the study of Hilbert schemes, since with their help it is

Journal of Symbolic Computation, 1996

In this paper we will define analogs of Gröbner bases for R-subalgebras and their ideals in a polynomial ring R[x 1 ,. .. , xn] where R is a noetherian integral domain with multiplicative identity and in which we can determine ideal membership and compute syzygies. The main goal is to present and verify algorithms for constructing these Gröbner basis counterparts. As an application, we will produce a method for computing generators for the first syzygy module of a subset of an R-subalgebra of R[x 1 ,. .. , xn] where each coordinate of each syzygy must be an element of the subalgebra.

Journal of Symbolic Computation, 1991

In this paper a new notion of reduction depending on an arbitrary non-empty set ORD of term orderings on a polynomial ring is introduced. A general Buchberger algorithm based on this notion is devised. For a single element set ORD it specializes to the ordinary Buehberger algorithm. For ORD being the set of all term orderings a particular universal Gr~Sbner basis is constructed. We only deal with the ease K[x,y] since for higher dimensions we have not been able to prove that the generalized algorithm stops after a finite number of steps. Some reasons for understanding the underlying difticulties are given.

Beiträge Zur Algebra Und Geometrie / Contributions To Algebra And Geometry, 2020

The border basis scheme B O parametrizes all 0-dimensional ideals in K [x 1 ,. .. , x n ], where K is an arbitrary field, which have a border basis with respect to a given order ideal of terms O. Its vanishing ideal I (B O) is generated by quadratic equations which are easy to describe, and it contains a unique monomial point, namely the point corresponding to the monomial ideal generated by the terms outside O. Based on a detailed study of the generators of I (B O), we describe a K-basis of the cotangent space m/m 2 , where m is the maximal ideal of the monomial point. Moreover, we provide an efficient algorithm to compute such a basis and use it to characterize the regularity of the monomial point.

International Virtual FDP on “Frontiers of Mathematics”, 2020

* Introduction ? Varieties ? Ideals ? Linear Case ? Polynomials of One Variable * Gröbner Bases ? Term Orders ? S-Polynomials ? Buchberger’s Algorithm ? Sample Computations * Some Application of Gröbner Bases ? The 3-Color Problem ? Automatic Geometric Theorem Proving ? Other Applications * References

Papers from the international symposium on Symbolic and algebraic computation - ISSAC '92, 1992

Grobner bases are an important tool. Therefore the mainComputerAl gebraSy stems contain procedures for computing such ideal bases. These algorithms typically spend much time in reducing so called " S-polynomials" to O. One of Buchberger's two criteria for avoiding superfluous reductions to O was interpreted in paper by G ebauer & iVIoller [1988) as a criterion for finding in a, generating system of particular syzygies redundant ones. In the present paper this idea is extended. While the Grobner basis is constructed, a set of syzygies is updated and an " S-polynomial" is not considered if its coresponcling syzygy turns out to depend on the already known syzygies. Using this concept, we cover both of Buchberger's criteria, are able to include information on polynomials (i.e. syzygies) either obtained from input or from intermediate calculations, ancl avoid more superfluous reductions. We illustrate this procedure by two examples. * This reseal ch was made in the preliminary part of the research fiTI.Ln(. ecl with the CEC Basic Research ESPRIT contract n. 6846

Mathematics in Computer Science, 2020

The Border Basis Algorithm (BBA) still suffers from the lack of analogues of Buchberger's criteria for avoiding unnecessary reductions. In this paper we develop a signature based technique which provides a first remedial step: signature bounds allow us to recognize multiple reductions of the same ancestor polynomial. The new signature based algorithm is also combined with the Boolean BBA for ideals of Boolean polynomials. Experiments show that it is at least 5 times faster than the standard (Boolean) BBA.