Computing all border bases for ideals of points

Journal of Symbolic Computation, 2009

The Buchberger-Möller algorithm is a well-known efficient tool for computing the vanishing ideal of a finite set of points. If the coordinates of the points are (imprecise) measured data, the resulting Gröbner basis is numerically unstable. In this paper we introduce a numerically stable Approximate Vanishing Ideal (AVI) Algorithm which computes a set of polynomials that almost vanish at the given points and almost form a border basis. Moreover, we provide a modification of this algorithm which produces a Macaulay basis of an approximate vanishing ideal. We also generalize the Border Basis Algorithm ([Kehrein, A., Kreuzer, M., 2006. Computing border bases. J. Pure Appl. Algebra 205, 279-295]) to the approximate setting and study the approximate membership problem for zero-dimensional polynomial ideals. The algorithms are then applied to actual industrial problems.

Journal of Symbolic Computation, 2000

We address the problem of computing ideals of polynomials which vanish at a finite set of points. In particular we develop a modular Buchberger-Möller algorithm, best suited for the computation over Q, and study its complexity; then we describe a variant for the computation of ideals of projective points, which uses a direct approach and a new stopping criterion. The described algorithms are implemented in CoCoA, and we report some experimental timings.

2007

We present an algorithm for computing Gröbner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies a set of essential variables, which reduces the time complexity with respect to the number of indeterminates, and then uses PLU decompositions to reduce the time complexity with respect to the number of points. This gives a theoretical upper bound for its time complexity that is an order of magnitude lower than the known one for the standard Buchberger-Möller algorithm if the number of indeterminates is much larger than the number of points. Comparison of implementations of our algorithm and the standard Buchberger-Möller algorithm in Macaulay 2 confirm the theoretically predicted speedup. This work is motivated by recent applications of Gröbner bases to the problem of network reconstruction in molecular biology.

Journal of Symbolic Computation, 2011

We construct an explicit minimal strong Gröbner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m ≥ 2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Gröbner basis is independent of the monomial order and that the set of leading terms of the constructed Gröbner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Gröbner basis in Z/m[x1, x2, . . . , xn] along the prime factorization of m. The obtained results are not only of mathematical interest but have immediate applications in formal verification of data paths for microelectronic systems-on-chip.

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.

2006

Grobner basis theory is a fundamental tool of computational com- mutative algebra. The theory has been advanced by the introduction of techniques from combinatorics, polyhedral geometry and computational geometry. In particular, such techniques were used to create the con- cept of Grobner region for an ideal of a polynomial ring. The purpose of this paper is to present algoritms for computing the Grobner region of a principal ideal in two indeterminates, to implement in Singular (1) and also to visualize this object with Mathematica (2).

ACM Communications in Computer Algebra, 2006

A contemporary and exciting application of Gröbner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to tens of points, while the number of genes or variables is potentially in the thousands. As such data sets vastly underdetermine the biological network, many models may fit the same data and reverse engineering programs often require the use of methods for choosing parsimonious models. Gröbner bases have recently been employed as a selection tool for polynomial dynamical systems that are characterized by maps in a vector space over a finite field. While there are numerous existing algorithms to compute Gröbner bases, to date none has been specifically designed to cope with large numbers of variables and few distinct data points. In this paper, we present an algorithm for computing Gröbner bases of zero-dimensional ideals that is optimized for the ca...

2007

We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner basis is independent of the monomial order and that the set of leading terms of the constructed Groebner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Groebner basis in Z/m[x_1,x_2,...,x_n] along the prime factorization of m. The obtained results are not only of mathematical interest but have immediate applications in formal verification of data paths for microelectronic systems-on-chip.

Journal of Symbolic Computation, 2019

We provide an algorithm that allows to describe the minimal log-resolution of an ideal in a smooth complex surface from the minimal log-resolution of its generators. In order to make this algorithm effective we present a modified version of the Newton-Puiseux algorithm that allows to compute the Puiseux decomposition of a product of not necessarily reduced or irreducible elements together with their algebraic multiplicity in each factor.

Proceedings of the twenty-first international symposium on Symbolic and algebraic computation - ISSAC '08, 2008

A symmetric ideal I ⊆ R = K[x 1 , x 2 , . . .] is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gröbner bases for symmetric ideals in the infinite dimensional polynomial ring R. This allows for symbolic computation in a new class of rings. In particular, we solve the ideal membership problem for symmetric ideals of R.