Zero-dimensional Ideals Research Articles

On Saturation of Zero-Dimensional Ideals

On Saturation of Zero-Dimensional Ideals

Effective algorithm for computing Noetherian operators of zero-dimensional ideals

Effective algorithm for computing Noetherian operators of zero-dimensional ideals

Asymptotic multiplicities and Monge–Ampère masses (with an appendix by Sébastien Boucksom)

Ein, Lazarsfeld and Smith asked whether ‘equality’ holds between two Samuel type asymptotic multiplicities for a graded system of zero-dimensional ideals on a smooth complex variety. We find a connection of this question to complex analysis by showing that the ‘equality’ is equivalent to a particular case of Demailly’s strong continuity property on the convergence of residual Monge–Ampère masses under approximation of plurisubharmonic functions. On the other hand, in an appendix of this paper, Sébastien Boucksom gives an algebraic proof of the ‘equality’ in general, using the intersection theory of b-divisors. We then use these to show that Demailly’s strong continuity holds for a new important class of plurisubharmonic functions.

Computing PUR of Zero-Dimensional Ideals of Breadth at Most One

Computing PUR of Zero-Dimensional Ideals of Breadth at Most One

Canonical Hilbert-Burch matrices for power series

Sets of zero-dimensional ideals in the polynomial ring k[x,y] that share the same leading term ideal with respect to a given term ordering are known to be affine spaces called Gröbner cells. Conca-Valla and Constantinescu parametrize such Gröbner cells in terms of certain canonical Hilbert-Burch matrices for the lexicographical and degree-lexicographical term orderings, respectively.In this paper, we give a parametrization of (x,y)-primary ideals in Gröbner cells which is compatible with the local structure of such ideals. More precisely, we extend previous results to the local setting by defining a notion of canonical Hilbert-Burch matrices of zero-dimensional ideals in the power series ring k〚x,y〛 with a given leading term ideal with respect to a local term ordering.

Computing and using minimal polynomials

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I. It is well known that minimal polynomials may be computed via elimination, therefore this is considered to be a “resolved problem”. But being the key of so many computations, it is worth investigating its meaning, its optimization, its applications (e.g. testing if a zero-dimensional ideal is radical, primary or maximal). We present efficient algorithms for computing the minimal polynomial of an element of P/I. For the specific case where the coefficients are in Q, we show how to use modular methods to obtain a guaranteed result. We also present some applications of minimal polynomials, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality.

On W-characteristic sets of lexicographic Gröbner bases

The structures of lexicographic (LEX) Gröbner bases were studied first by Lazard [4] for bivariate ideals and then extended to general zero-dimensional multivariate (radical) ideals [3, 6, 2]. Based on the structures of LEX Gröbner bases, algorithms have been proposed to compute triangular decompositions out of LEX Gröbner bases for zero-dimensional ideals [5, 2]. The relationships between LEX Gröbner bases and Ritt characteristic sets were explored in [1] and then made clearer in [8] with the concept of W-characteristic sets.

An algorithm for computing the Hilbert–Samuel multiplicities and reductions of zero-dimensional ideals of Cohen–Macaulay local rings

In this paper, we present an algorithm for computing the minimal reductions of m-primary ideals of Cohen–Macaulay local rings. Using this algorithm, we are able to compute the Hilbert–Samuel multiplicities and solve the membership problem for the integral closure of m-primary ideals.

Solving Parametric Ideal Membership Problems and Computing Integral Numbers in a Ring of Convergent Power Series Via Comprehensive Gröbner Systems

New algorithms are presented for solving ideal membership problems with parameters and for computing integral numbers in a ring of convergent power series. It is shown that the ideal membership problems for zero-dimensional ideals in the ring of convergent power series, can be solved in the polynomial ring. The key idea of the algorithms is computing comprehensive Grobner systems of ideal quotients in a polynomial ring. Furthermore, new methods for computing integral numbers, in the local ring, are introduced as an application.

Algebraic local cohomology with parameters and parametric standard bases for zero-dimensional ideals

A computation method of algebraic local cohomology classes, associated with zero-dimensional ideals with parameters, is introduced. This computation method gives us in particular a decomposition of the parameter space depending on the structure of algebraic local cohomology classes. This decomposition informs us on several properties of input ideals and the output of the proposed algorithm completely describes the multiplicity structure of input ideals. An algorithm for computing a parametric standard basis of a given zero-dimensional ideal, with respect to an arbitrary local term order, is also described as an application of the computation method. The algorithm can always output “reduced” standard basis of a given zero-dimensional ideal, even if the zero-dimensional ideal has parameters.

On the functoriality of marked families

The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow associating to each ideal $J$ of this type a scheme $\MFScheme {J}$, called a $J$-marked scheme. In this paper, we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not depend on the ring of coefficients. We prove that, for all strongly stable ideals $J$, the marked schemes $\MFScheme {J}$ can be embedded in a Hilbert scheme as locally closed subschemes, and that they are open under suitable conditions on $J$. Finally, we generalize Lederer's result about Gr\"obner strata of zero-dimensional ideals, proving that Gr\"obner strata of any ideals are locally closed subschemes of Hilbert schemes.

An algebraic approach to the scattering equations

We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

On real tropical bases and real tropical discriminants

We explore the concept of real tropical basis of an ideal in the field of real Puiseux series. We explicitly construct tropical bases of zero-dimensional real radical ideals, linear ideals and hypersurfaces coming from combinatorial patchworking. But we also show that there exist real radical ideals that do not admit a tropical basis. As an application, we show how to compute the set of singular points of a real tropical hypersurface, i.e. we compute the real tropical discriminant.

NUMERICAL ALGORITHMS FOR DUAL BASES OF POSITIVE-DIMENSIONAL IDEALS

An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However the usual standard basis algorithms are not numerically stable. A numerically stable approach to describing the ideal is by finding the space of dual functionals that annihilate it, which reduces the problem to one of linear algebra. There are several known algorithms for finding the truncated dual up to any specified degree, which is useful for describing zero-dimensional ideals. We present a stopping criterion for positive-dimensional cases based on homogenization that guarantees all generators of the initial monomial ideal are found. This has applications for calculating Hilbert functions.

Zero-dimensional ideals of limited precision points

Zero-dimensional ideals of limited precision points

Finite Fields, Gröbner Bases and Modular Secret Sharing

We propose a new application of the Gröbner basis methods. In particular, we construct generalized modular secret sharing. Basically, the secret sharing is the pair of splitting and restoring algorithms. Initially, the secret key is divided into the shares containing partial information of the secret so that only specified sets of participants can restore it pooling their subsets of shares together. We provide the implementation of both algorithms of secret sharing using Gröbner basis methods. Therefore, we call the proposed construction GB secret sharing. The most important feature of our approach is that the original modular construction in the ring of integers and polynomial ring Fq [x] over the Galois field Fq is generalized to the multivariate polynomial ring Fq [x 1, x 2, ‥, xn ]. We use the ideals defined by their Gröbner bases in the place of integer and polynomial moduli. First, we prove that any access structure possesses a GB realization in any multivariate polynomial ring Fq [x 1, x 2,‥, xn ]. This is the generalization of our previous result in the ring of polynomials over the Galois field Fq and the results of Mignotte, Asmuth and Bloom for the threshold access structures in the ring of integers. Second, we give the characterization and count the special zero-dimensional ideals in Fq [x 1, x 2,‥, xn ] which are necessary for the realization of GB secret sharing.

Computing polynomial univariate representations of zero-dimensional ideals by Gröbner basis

Rational Univariate Representation (RUR) of zero-dimensional ideals is used to describe the zeros of zero-dimensional ideals and RUR has been studied extensively. In 1999, Roullier proposed an efficient algorithm to compute RUR of zero-dimensional ideals. In this paper, we will present a new algorithm to compute Polynomial Univariate Representation (PUR) of zero-dimensional ideals. The new algorithm is based on some interesting properties of Grobner basis. The new algorithm also provides a method for testing separating elements.

FINITE SETS OF AFFINE POINTS WITH UNIQUE ASSOCIATED MONOMIAL ORDER QUOTIENT BASES

The quotient bases for zero-dimensional ideals are often of interest in the investigation of multivariate polynomial interpolation, algebraic coding theory, and computational molecular biology, etc. In this paper, we discuss the properties of zero-dimensional ideals with unique monomial order quotient bases, and verify that the vanishing ideals of Cartesian sets have unique monomial order quotient bases. Furthermore, we reveal the relation between Cartesian sets and the point sets with unique associated monomial order quotient bases.

Improved decoding of affine-variety codes

General error locator polynomials are polynomials able to decode any correctable syndrome for a given linear code. Such polynomials are known to exist for all cyclic codes and for a large class of linear codes. We provide some decoding techniques for affine-variety codes using some multidimensional extensions of general error locator polynomials. We prove the existence of such polynomials for any correctable affine-variety code and hence for any linear code. We propose two main different approaches, that depend on the underlying geometry. We compute some interesting cases, including Hermitian codes. To prove our coding theory results, we develop a theory for special classes of zero-dimensional ideals, that can be considered generalizations of stratified ideals. Our improvement with respect to stratified ideals is twofold: we generalize from one variable to many variables and we introduce points with multiplicities.

Multiplicity bounds in graded rings

The F-threshold cJ(a) of an ideal a with respect to an ideal J is a positive characteristic invariant obtained by comparing the powers of a with the Frobenius powers of J. We study a conjecture formulated in an earlier article that we authored with M. Mustaţă, which bounds cJ(a) in terms of the multiplicities e(a) and e(J) when a and J are zero-dimensional ideals and J is generated by a system of parameters. We prove the conjecture when a and J are generated by homogeneous systems of parameters in a Noetherian graded k-algebra. We also prove a similar inequality involving, instead of the F-threshold, the jumping number for the generalized parameter test submodules.