Monomial Order Research Articles

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.

Software for computing the Gröbner cover of a parametric ideal

demonstration Share on Software for computing the Gröbner cover of a parametric ideal Author: Antonio Montes Universitat Politàcnica de Catalunya, Spain Universitat Politàcnica de Catalunya, SpainView Profile Authors Info & Claims ACM Communications in Computer AlgebraVolume 45Issue 3/4September/December 2011 pp 180–182https://doi.org/10.1145/2110170.2110178Published:23 January 2012Publication History 0citation48DownloadsMetricsTotal Citations0Total Downloads48Last 12 Months0Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Dimension-dependent bounds for Gröbner bases of polynomial ideals

Given a basis F of a polynomial ideal I in K[x1,…,xn] with degrees deg(F)≤d, the degrees of the reduced Gröbner basis G w.r.t. any admissible monomial ordering are known to be double exponential in the number of indeterminates in the worst case, i.e. deg(G)=d2Θ(n). This was established in Mayr and Meyer (1982) andDubé (1990).We modify both constructions in order to give worst case bounds depending on the ideal dimension proving that deg(G)=dnΘ(1)2Θ(r) for r-dimensional ideals (in the worst case).

Subalgebra Analogue to standard basis for ideal

A theory of “subalgebra basis” analogous to standard basis (the generalization of Gröbner bases to monomial orderings which are not necessarily well orderings [1]) for ideals in polynomial rings over a field is developed. We call these bases “SASBI Basis” for “Subalgebra Analogue to Standard Basis for Ideals”. The case of global orderings, here they are called “SAGBI Basis” for “Subalgebra Analogue to Gröbner Basis for Ideals”, is treated in [6]. Sasbi bases may be infinite. In this paper we consider subalgebras admitting a finite Sasbi basis and give algorithms to compute them.

Archimedean Orders on Certain Rings of Invariants

Let [Formula: see text] act multiplicatively on the Laurent polynomial algebra k[x± 1] in n indeterminates x={x1, …,xn}. Consider the initial algebra of the ring of invariants [Formula: see text] with respect to some monomial order. We set a sufficient condition on [Formula: see text] such that each initial algebra is represented by some weight vectors in ℝn. We also show that the condition is necessary in the case where the rank n = 2.

Toric ideals of lattice path matroids and polymatroids

White has conjectured that the toric ideal of a matroid is generated by quadric binomials corresponding to symmetric basis exchanges. We prove a stronger version of this conjecture for lattice path polymatroids by constructing a monomial order under which these sets of quadrics form Gröbner bases. We then introduce a larger class of polymatroids for which an analogous theorem holds. Finally, we obtain the same result for lattice path matroids as a corollary.

Edge ideals and connection problems

Ideals arising from graphs are investigated via Grobner bases theory in order to introduce algebraic objects useful for applications related to the field of security. In particular, the notion of s-sequence for the generators of the edge ideal I(G) of any graph G is considered, so that a description of the Grobner basis for the relation ideal J of the symmetric algebra of I(G) can be obtained. Based on this approach, the initial ideal of J with respect to a monomial order is well-determined and defines the edge ideal of a supporting graph F, essential in transmitting.

The Gröbner basis of the ideal of vanishing polynomials

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 [ 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.

Strong non-Noetherity of polynomial reduction

It is a well-known result by A. Reeves and B. Sturmfels that the reduction modulo a marked set of polynomials is Noetherian if and only if the marking is induced from an admissible term ordering. For finite sets of polynomials with a nonadmissible ordering, there is a constructive proof of the existence of an infinite reduction sequence, although a finite one might still be possible. On the base of our specialized software for combinatorics of monomial orderings, we have found some examples for which there is no finite reduction sequence. This is what we call “strong” non-Noetherity. Bibliography: 3 titles.

Normalization of rings

We present a new algorithm to compute the integral closure of a reduced Noetherian ring in its total ring of fractions. A modification, applicable in positive characteristic, where actually all computations are over the original ring, is also described. The new algorithm of this paper has been implemented in Singular, for localizations of affine rings with respect to arbitrary monomial orderings. Benchmark tests show that it is in general much faster than any other implementation of normalization algorithms known to us.

Computation of Non-Commutative Gröbner Bases in Grassmann and Clifford Algebras

Tensor, Clifford and Grassmann algebras belong to a wide class of non-commutative algebras that have a Poincare-Birkhoff-Witt (PBW) “monomial” basis. The necessary and sufficient condition for an algebra to have the PBW basis has been established by T. Mora and then V. Levandovskyy as the so called “non-degeneracy condition”. This has led V. Levandovskyy to a re-discovery of the so called G-algebras (previously introduced by J. Apel) and GR-algebras (Grobner-ready algebras). It was T. Mora who already in the 1990s considered a comprehensive and algorithmic approach to Grobner bases for commutative and non-commutative algebras. It was T. Stokes who eighteen years ago introduced Grobner left bases (GLB) and Grobner left ideal bases (GLIB) for left ideals in Grassmann algebras, with the GLIB bases solving an ideal membership problem. Thus, a natural question is to first seek Grobner bases with respect to a suitable admissible monomial order for ideals in tensor algebras T and then consider quotient algebras T/I. It was shown by Levandovskyy that these quotient algebras possess a PBW basis if and only if the ideal I has a Grobner basis. Of course, these quotient algebras are of great interest because, in particular, Grassmann and Clifford algebras of a quadratic form arise this way. Examples of G-algebras include the quantum plane, universal enveloping algebras of finite dimensional Lie algebras, some Ore extensions, Weyl algebras and their quantizations, etc. Examples of GR-algebras, which are either G algebras or are isomorphic to quotient algebras of a G-algebra modulo a proper two-sided ideal, include Grassmann and Clifford algebras. After recalling basic concepts behind the theory of commutative Grobner bases, a review of the Grobner bases in PBW algebras, G-,and GR-algebras will be given with a special emphasis on computation of such bases in Grassmann and Clifford algebras. GLB and GLIB bases will also be computed.

On computation of Boolean involutive bases

Grobner bases in Boolean rings can be calculated by an involutive algorithm based on the Janet or Pommaret division. The Pommaret division allows calculations immediately in the Boolean ring, whereas the Janet division implies use of a polynomial ring over field $$ \mathbb{F}_2 $$ . In this paper, both divisions are considered, and distributive and recursive representations of Boolean polynomials are compared from the point of view of calculation effectiveness. Results of computer experiments with both representations for an algorithm based on the Pommaret division and for lexicographical monomial order are presented.

Design of quadrature rules for Müntz and Müntz‐logarithmic polynomials using monomial transformation

AbstractA method for constructing the exact quadratures for Müntz and Müntz‐logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of Müntz‐logarithmic polynomials in terms of the number of Gauss–Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end‐point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run‐time evaluation than the existing methods for Müntz polynomials. Copyright © 2009 John Wiley & Sons, Ltd.

Hilbert functions of points on Schubert varieties in orthogonal Grassmannians

Given a point on a Schubert variety in an orthogonal Grassmannian, we compute the multiplicity, more generally the Hilbert function. We first translate the problem from geometry to combinatorics by applying standard monomial theory. The solution of the resulting combinatorial problem forms the bulk of the paper. This approach has been followed earlier to solve the same problem for Grassmannians and symplectic Grassmannians. As an application, we present an interpretation of the multiplicity as the number of non-intersecting lattice paths of a certain kind. A more important application, although it does not appear here but elsewhere, is to the computation of the initial ideal, with respect to certain convenient monomial orders, of the ideal of the tangent cone to the Schubert variety. Taking the Schubert variety to be of a special kind and the point to be the `identity coset,' our problem specializes to one about Pfaffian ideals, treatments of which by different methods exist in the literature. Also available in the literature is a geometric solution when the point is a `generic singularity.'

Nullstellensätze for Zero–Dimensional Gröbner Bases

In this paper, we give first some non-trivial improvements of the well-known bounds of effective Nullstellensatze. Using these bounds, we show that the Grobner basis (for any monomial ordering) of a zero–dimensional ideal may be computed within a bit complexity which is essentially polynomial in $${D^{n^{2}}}$$where n is the number of unknowns and D is the mean value of the degrees of input polynomials.

Enumeration of finite monomial orderings and combinatorics of universal Gröbner bases

The goal of this work is to analyze various classes of finite and total monomial orderings. The concept of monomial ordering plays the key role in the theory of Grobner bases: every basis is determined by a certain ordering. At the same time, in order to define a Grobner basis, it is not necessary to know ordering of all monomials. Instead, it is sufficient to know only a finite interval of the given ordering. We consider combinatorics of finite monomial orderings and its relationship with cells of a universal Grobner basis. For every considered class of orderings (weakly admissible, convex, and admissible), an algorithm for enumerating finite orderings is discussed and combinatorial integer sequences are obtained. An algorithm for computing all minimal finite orderings for an arbitrary Grobner basis that completely determine this basis is presented. The paper presents also an algorithm for computing an extended universal Grobner basis for an arbitrary zero-dimensional ideal.

Gröbner deformations, connectedness and cohomological dimension

In this paper we will compare the connectivity dimension c ( P / I ) of an ideal I in a polynomial ring P with that of any initial ideal of I. Generalizing a theorem of Kalkbrener and Sturmfels [M. Kalkbrener, B. Sturmfels, Initial complex of prime ideals, Adv. Math. 116 (1995) 365–376], we prove that c ( P / LT ≺ ( I ) ) ⩾ min { c ( P / I ) , dim ( P / I ) − 1 } for each monomial order ≺. As a corollary we have that every initial complex of a Cohen–Macaulay ideal is strongly connected. Our approach is based on the study of the cohomological dimension of an ideal a in a noetherian ring R and its relation with the connectivity dimension of R / a . In particular we prove a generalized version of a theorem of Grothendieck [A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), in: Séminaire de Géométrie Algébrique du Bois Marie, 1962]. As consequence of these results we obtain some necessary conditions for an open subscheme of a projective scheme to be affine.

Involutive method for computing Gröbner bases over % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv39ga % iyaacqWFfcVrdaWgaaWcbaGaeGOmaidabeaaaaa!47CE! $$ \mathbb{F}_2 $$

In this paper, an involutive algorithm for computation of Grobner bases for polynomial ideals in a ring of polynomials in many variables over the finite field $$ \mathbb{F}_2 $$ with the values of variables belonging of $$ \mathbb{F}_2 $$ is considered. The algorithm uses Janet division and is specialized for a graded reverse lexicographical order of monomials. We compare efficiency of this algorithm and its implementation in C++ with that of the Buchberger algorithm, as well as with the algorithms of computation of Grobner bases that are built in the computer algebra systems Singular and CoCoA and in the FGb library for Maple. For the sake of comparison, we took widely used examples of computation of Grobner bases over ? and adapted them for $$ \mathbb{F}_2 $$ . Polynomial systems over $$ \mathbb{F}_2 $$ with the values of variables in $$ \mathbb{F}_2 $$ are of interest, in particular, for modeling quantum computation and a number of cryptanalysis problems.

An Eigenvalue Method for Testing Positive Definiteness of a Multivariate Form

In this paper, we present an eigenvalue method for testing positive definiteness of a multivariate form. This problem plays an important role in the stability study of nonlinear autonomous systems via Lyapunov's direct method in automatic control. At first we apply the D'Andrea-Dickenstein version of the classical Macaulay formulas of the resultant to compute the symmetric hyperdeterminant of an even order supersymmetric tensor. By using the supersymmetry property, we give detailed computation procedures for the Bezoutians and specified ordering of monomials in this approach. We then use these formulas to calculate the characteristic polynomial of a fourth order three dimensional supersymmetric tensor and give an eigenvalue method for testing positive definiteness of a quartic form of three variables. Some numerical results of this method are reported.

Standard bases in [formula omitted

In this paper we study standard bases for submodules of K [ [ t 1 , … , t m ] ] [ x 1 , … , x n ] s respectively of their localisation with respect to a t ¯ -local monomial ordering. The main step is to prove the existence of a division with remainder generalising and combining the division theorems of Grauert–Hironaka and Mora. Everything else then translates naturally. Setting either m = 0 or n = 0 we get standard bases for polynomial rings respectively for power series rings as a special case. We then apply this technique to show that the t -initial ideal of an ideal over the Puiseux series field can be read of from a standard basis of its generators. This is an important step in the constructive proof that each point in the tropical variety of such an ideal admits a lifting.