An algebraist’s view on border bases

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 ...

Journal of Pure and Applied Algebra, 2005

This paper presents characterizations of border bases of zero-dimensional polynomial ideals that are analogous to the known characterizations of Gröbner bases. Based on a Border Division Algorithm, a variant of the usual Division Algorithm, we characterize border bases as border prebases with one of the following equivalent properties: special generation, generation of the border form ideal, confluence of the corresponding rewrite relation, reduction of S-polynomials to zero, and lifting of syzygies. The last characterization relies on a detailed study of the relative position of the border terms and their syzygy module. In particular, a border prebasis is a border basis if and only if all fundamental syzygies of neighboring border terms lift; these liftings are easy to compute.

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

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.

Journal of Algebra, 2007

Let (S, n) be a 2-dimensional regular local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Let I * be the leading ideal of I in the associated graded ring gr n (S), and set R = S/I and m = n/I. In [GHK2], we prove that if µ G (I *) = n, then I * contains a homogeneous system {ξ i } 1≤i≤n of generators such that deg ξ i + 2 ≤ deg ξ i+1 for 2 ≤ i ≤ n−1, and htG(ξ1, ξ2, • • • , ξn−1) = 1, and we describe precisely the Hilbert series H(gr m (R), λ) in terms of the degrees c i of the ξ i and the integers d i , where di is the degree of Di = GCD(ξ1,. .. , ξi). To the complete intersection ideal I = (f, g)S we associate a positive integer n with 2 ≤ n ≤ c 1 + 1, an ascending sequence of positive integers (c 1 , c 2 ,. .. , c n), and a descending sequence of integers (d1 = c1, d2,. .. , dn = 0) such that ci+1 − ci > di−1 − di > 0 for each i with 2 ≤ i ≤ n − 1. We establish here that this necessary condition is also sufficient for there to exist a complete intersection ideal I = (f, g) whose leading ideal has these invariants. We give several examples to illustrate our theorems.

Mathematics of Computation, 2007

In this paper we study the structure of Gröbner bases with respect to block orders. We extend Lazard's theorem and the Gianni-Kalkbrenner theorem to the case of a zero-dimensional ideal whose trace in the ring generated by the first block of variables is radical. We then show that they do not hold for general zero-dimensional ideals.

2005

The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal (M) generated by the boundaries of the circuits of the matroid. There is an isomorphism between the Orlik-Solomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon algebras, called χ-algebras, are considered. These new algebras include, apart from the Orlik-Solomon algebras, the Orlik-Solomon-Terao algebra of a set of vectors and the Cordovil algebra of an oriented matroid. To encode an important property of the "no broken circuit bases" of the Orlik-Solomon-Terao algebras, András Szenes has introduced a particular type of bases, the so called "diagonal bases". This notion extends naturally to the χ-algebras. We give a survey of the results obtained by the authors concerning the construction of Gröbner bases of χ(M) and diagonal bases of Orlik-Solomon type algebras and we present the combinatorial analogue of an "iterative residue formula" introduced by Szenes.

Advances in Mathematics, 1992

2004