Computing Ideals of Points

Journal of Symbolic Computation, 2000

We present an algorithm for computing a minimal set of generators for the ideal of a rational parametric projective curve in polynomial time. The method exploits the availability of polynomial algorithms for the computation of minimal generators of an ideal of points and is an alternative to the existing Gröbner bases techniques for the implicitization of curves. The termination criterion is based on the Castelnuovo-Mumford regularity of a curve. The described computation also yields the Hilbert function and, hence, the Hilbert polynomial and the Poincaré series of the curves. Moreover, it can be applied to unions of rational curves. We have compared the implementation of our algorithm with the Hilbert driven elimination algorithm included in CoCoA 3.6 and Singular 1.2, obtaining, in general, significant improvements in 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, 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.

Proceedings of the 1998 international symposium on Symbolic and algebraic computation, 1998

In this paper we review the known algorithms for performing the basic algorithms for ideal and submodule operations: intersection, transporter and saturation. The algorithms known in the literature for these operations on polynomial rings fall largely into two classes: syzygy algorithms and elimination algorithms. We show that the two classes substantially coincide: they can be seen at most as variants of the same algorithm. We show moreover that these algorithms can be generalized to another algorithm, a module elimination algorithm, that allows the use of a Hilbert function driven algorithm, see Tr , and that, with this feature, appears to be the most e cient algorithm in this class. We give some examples that support this assertion. Because of space constraints we skip all the proofs, that will appear in a full paper together with more exhaustive experiments.

Journal of Symbolic Computation, 2000

This paper presents an algorithm for the Quillen-Suslin Theorem for quotients of polynomial rings by monomial ideals, that is, quotients of the form A = k x 0 ; :::;xn]=I, with I a monomial ideal and k a eld. T. Vorst proved that nitely generated projective modules over such algebras are free. Given a nitely generated module P, described by generators and relations, the algorithm tests whether P is projective, in which case it computes a free basis for P.

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 Complexity, 1997

In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)-tuple P = ( f, g 1 , g 2 , . . . , g w ) where f and the g i are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert's Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases. © 1997 Academic Press

Journal of Algebra, 1983

Let C be an irreducible projective curve of degree d in Pn(K), where K is an algebraically closed field, and let I be the associated homogeneous prime ideal. We wish to compute generators for I, assuming we are given sufficiently many points on the curve C. In particular if I can be generated by polynomials of degree at most m and we are given md + 1 points on C, then we can find a set of generators for I. We will show that a minimal set of generators of I can be constructed in polynomial time. Our constructions are completely independent of any notion of term ordering; this allows us the maximal freedom in performing our constructions in order to improve the numerical stability. We also summarize some classical results on bounds for the degrees of the generators of our ideal in terms of the degree and genus of the curve.