Zero-dimensional Ideals Research Articles

On border basis and Gröbner basis schemes

Hilbert schemes of zerodimensional ideals in a polynomial ring can be covered with suitable affine open subschemeswhose 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 Grobner bases which, nevertheless, can also be used in the study of Hilbert schemes, since they provide tools for constructing suitable stratifications.

On the upper bound of the multiplicity conjecture

Let A = K[X 1 ,..., X n ] and let I be a graded ideal in A. We show that the upper bound of the multiplicity conjecture of Herzog, Huneke and Srinivasan holds asymptotically (i.e., for I k and all k » 0) if I belongs to any of the following large classes of ideals: (1) radical ideals, (2) monomial ideals with generators in different degrees, (3) zero-dimensional ideals with generators in different degrees. Surprisingly, our proof uses local techniques like analyticity, reductions, equimultiplicity and local results like Rees's theorem on multiplicities.

Primary decomposition of zero-dimensional ideals over finite fields

Gao, Shuhong Clemson Univ, Dept Math Sci, Clemson, SC 29634 USA. Wan, Daqing Univ Calif Irvine, Dept Math, Irvine, CA 92697 USA. Wang, Mingsheng Acad Sinica, Inst Software, Informat Secur Lab, Beijing 100080, Peoples R China.

Interpolation in list decoding of Reed-Solomon codes

We consider the problem of efficient implementation of two-dimensional interpolation in the Guruswami-Sudan list decoding algorithm for Reed-Solomon codes. We show that it can be implemented by computing the product of ideals of interpolation polynomials constructed for subsets of interpolation points. A method for fast multiplication of coprime zero-dimensional ideals is proposed.

Structure of Gröbner bases with respect to block orders

In this paper we study the structure of Grobner 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.

Computing Gröbner bases of ideals of few points in high dimensions

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 case when the number m of points is much smaller than the number n of indeterminates. The algorithm identifies those variables that are essential , that is, in the support of the standard monomials associated to a polynomial ideal, and computes the relations in the Gröbner basis in terms of these variables. When n is much larger than m , the complexity is dominated by nm 3 . The algorithm has been implemented and tested in the computer algebra system Macaulay 2 . We provide a comparison of its performance to the Buchberger-Möller algorithm, as built into the system.

On inverse systems and squarefree decomposition of zero-dimensional polynomial ideals

Let I be a zero-dimensional ideal in a polynomial ring F [ s ] : = F [ s 1 , … , s n ] over an arbitrary field F . We show how to compute an F -basis of the inverse system I ⊥ of I . We describe the F [ s ] -module I ⊥ by generators and relations and characterise the minimal length of a system of F [ s ] -generators of I ⊥ . If the primary decomposition of I is known, such a system can be computed. Finally we generalise the well-known notion of squarefree decomposition of a univariate polynomial to the case of zero-dimensional ideals in F [ s ] and present an algorithm to compute this decomposition.

Gröbner bases of certain zero-dimensional ideals arising in coding theory

The combinatorics of reduced Gröbner bases of certain zero-dimensional ideals, which arise when Gröbner basis technique is applied to the soft-decision maximum likelihood decoding of binary linear block codes, will be studied.

Decomposition of tableaus annihilated by zero-dimensional ideals

A (1-dimensional) linear recursive sequence s over a field K decomposes in the following canonical way. Let I⊂ K[ x] be the annihilator ideal of s. Since K[ x] is a principal ideal domain, I= fK[ x] for some polynomial f, which can be factored as f 1⋯ f r where the f i are coprime. Thus s can be uniquely written as a sum of sequences s i having annihilator ideals I i = f i K[ x]. Furthermore, each f i is a power of an irreducible polynomial f i =( g i ) e i . Each sequence s i can be uniquely written as a e i -fold sum of pointwise products of a “binomial” sequence with a sequence annihilated by g i . Finally, a sequence annihilated by an irreducible polynomial g i is given by a trace formula. See, for instance, [N. Zierler, W.H. Mills, J. Algebra 27 (1973) 147–157]. We show that a completely analogous decomposition (which subsumes the 1-dimensional case) holds for n-dimensional linear recursive sequences, i.e., tableaus annihilated by zero-dimensional ideals of K[ x 1,…, x n ].

PROPERTIES OF COMMUTATIVE ASSOCIATION SCHEMES DERIVED BY FGLM TECHNIQUES

Association schemes are combinatorial objects that allow us solve problems in several branches of mathematics. They have been used in the study of permutation groups and graphs and also in the design of experiments, coding theory, partition designs etc. In this paper we show some techniques for computing properties of association schemes. The main framework arises from the fact that we can characterize completely the Bose–Mesner algebra in terms of a zero-dimensional ideal. A Gröbner basis of this ideal can be easily derived without the use of Buchberger algorithm in an efficient way. From this statement, some nice relations arise between the treatment of zero-dimensional ideals by reordering techniques (FGLM techniques) and some properties of the schemes such as P-polynomiality, and minimal generators of the algebra.

Computing the Primary Decomposition of Zero-dimensional Ideals

Let K be an infinite perfect computable field and let I⊆K [ x ] be a zero-dimensional ideal represented by a Gröbner basis. We derive a new algorithm for computing the reduced primary decomposition of I using only standard linear algebra and univariate polynomial factorization techniques. In practice, the algorithm generally works in finite fields of large characteristic as well.

On multiplicities of graded sequences of ideals

We generalize a result from [L. Ein, et al., math. AG/0202303], proving that for an arbitrary graded sequence of zero-dimensional ideals, the multiplicity of the sequence is equal to its volume. This is done using a deformation to monomial ideals. As a consequence of our result, we obtain a formula which computes the multiplicity of an ideal I in terms of the multiplicities of the initial monomial ideals of the powers I m . We use this to give a new proof of the inequality between multiplicity and the log canonical threshold from [de Fernex, et al., J. Alg. Geom., to appear].

Multivariate Polynomial System Solving Using Intersections of Eigenspaces

The solutions of a polynomial system can be computed using eigenvalues and eigenvectors of certain endomorphisms. There are two different approaches, one by using the (right) eigenvectors of the representation matrices, one by using the (right) eigenvectors of their transposed ones, i.e. their left eigenvectors. For both approaches, we describe the common eigenspaces and give an algorithm for computing the solution of the algebraic system. As a byproduct, we present a new method for computing radicals of zero-dimensional ideals.

On the complexity of the real Nullstellensatz in the 0-dimensional case

We give single exponential bounds for the degrees in the identity of the Real Nullstellensatz for zero-dimensional ideals.

A Modular Method to Compute the Rational Univariate Representation of Zero-dimensional Ideals

To give an efficiently computable representation of the zeros of a zero-dimensional ideal I, Rouillier (1996) introduced the rational univariate representation (RUR) as an extension of the generalized shape lemma (GSL) proposed by Alonso et al. (1996). In this paper, we propose a new method to compute the RUR of the radical of I, and report on its practical implementation. In the new method, the RUR of the radical of I is computed efficiently by applying modular techniques to solving the systems of linear equations. The performance of the method is examined by practical experiments. We also discuss its theoretical efficiency.

Properties of Gröbner Bases and Applications to Doubly Periodic Arrays

In this paper the properties of Gröbner bases of zero-dimensional ideals are studied. A basis of the space of linear recurring arrays and the trace expression of linear recurring arrays are given.

Radical computations of zero-dimensional ideals and real root counting

Abstract The computation of the radical of a zero-dimensional ideal plays an important role in various areas of computer algebra. A bunch of different methods have been published to meet this task (see e.g. [4,8,11,13–17]). The method presented in this paper is strongly connected to a recent approach to the real root counting problem as described in [2,18]. It provides a lot of information for real root counting already in the process of calculating the radical. In this sense this approach is well-suited for real root counting problems.

On the Complexity of Sparse Elimination

Sparse elimination exploits the structure of a multivariate polynomial by considering its Newton polytope instead of its total degree. We concentrate on polynomial systems that generate zero-dimensional ideals. A monomial basis for the coordinate ring is defined from a mixed subdivision of the Minkowski sum of the Newton polytopes. We offer a new simple proof relying on the construction of a sparse resultant matrix, which leads to the computation of a multiplication map and all common zeros. The size of the monomial basis equals the mixed volume and its computation is equivalent to computing the mixed volume, so the latter is a measure of intrinsic complexity. On the other hand, our algorithms have worst-case complexity proportional to the volume of the Minkowski sum. In order to derive bounds in terms of the sparsity parameters, we establish new bounds on the Minkowski sum volume as a function of mixed volume. To this end, we prove a lower bound on mixed volume in terms of Euclidean volume which is of independent interest.

Gröbner and involutive bases for zero-dimensional ideals

A new characterization of involutive systems is presented, which leads to a connection between Gröbner basis and involutive systems, and allows to construct, in an easy way, an involutive basis from a Gröbner basis.

Determinantal formulae for the solution set of zero-dimensional ideals

In this paper we study the properties of a forgotten construction introduced by V.W. Habicht in 1948 related with the problem of computing a basis for the ideal generated by n polynomials F 1,…, F n in K[x n,…,x n allowing to determine in an easy way its zeros and its radical for the zero-dimensional case. This construction is achieved generalizing to K[x 1,…,x n the definition of subresultants of indexes 0 and 1 and so, the coefficients of the polynomials in the new basis are defined as the determinants of certain matrices constructed with the coefficients of the polynomials F 1,…, F n . The construction is introduced for the generic case and it is showed that its specialization works for all the zero-dimensional ideals in general position with respect to x n except when the coefficients of the polynomials F 1,…, F n are in some hypersurface in K n whose equation we shall give in an explicit way. A basis with a similar structure to the one here introduced can be computed using Groebner Basis (the Shape Lemma) imposing the radicality hypothesis to the initial ideal.