Standard bases concordant with the norm and computations in ideals and polylinear recurring sequences

Standard bases of ideals of the polynomial ring R[X] = R[x ₁, …, x _k ] over a commutative Artinian chain ring R that are concordant with the norm on R have been investigated by D. A. Mikhailov, A. A. Nechaev, and the author. In this paper we continue this investigation. We introduce a new order on terms and a new reduction algorithm, using the coordinate decomposition of elements from R. We prove that any ideal has a unique reduced (in terms of this algorithm) standard basis. We solve some classical computational problems: the construction of a set of coset representatives, the finding of a set of generators of the syzygy module, the evaluation of ideal quotients and intersections, and the elimination problem. We construct an algorithm testing the cyclicity of an LRS-family L _R (I), which is a generalization of known results to the multivariate case. We present new conditions determining whether a Ferre diagram $\mathcal{F}$ and a full system of $\mathcal{F}$ -monic polynomials form a shift register. On the basis of these results, we construct an algorithm for lifting a reduced Gröbner basis of a monic ideal to a standard basis with the same cardinality.     

Differential standard bases under composition

We generalize Hoon Hong’s theorem on Gröbner bases under composition to the case of differential standard bases in the ordinary ring of differential polynomials {ie4152-01}. In particular, we prove that some ideals have finite differential standard bases. We construct special orderings on differential monomials such that ideals generated by some power of a quasi-linear polynomial acquire finite differential standard bases.     

Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Gröbner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Gröbner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation.     

Candidates for nonzero Betti numbers of monomial ideals

Let I be a monomial ideal in the polynomial ring S generated by elements of degree at most d. In this paper, it is shown that, if the i-th syzygy of I has no elements of degrees j,…,j+(d?1) (where j≥i+d), then (i+1)-th syzygy of I does not have any element of degree j+d. Then we give several applications of this result, including an alternative proof for Green–Lazarsfeld index of the edge ideals of graphs as well as an alternative proof for Fröberg’s theorem on classification of square-free monomial ideals generated in degree 2 with linear resolution. Among all, we deduce a partial result on subadditivity of the syzygies for monomial ideals.     

Efficient polynomial reduction

H-bases are bases for polynomial ideals, characterized by the fact that their homogeneous leading terms are a basis for the associated homogeneous ideal. In the computation ofH-bases without term orders, an important task is to determine the orthogonal projection of a homogeneous polynomial to certain subspaces of homogeneous polynomials with respect to a given inner product. One way of doing so is to use an orthogonal basis of the subspace. In this paper, we present and study a method to efficiently compute such a basis for a particular but important inner product.     

Predicting syzygies over monomial relations algebras

We show that all projective resolutions over a monomial relations algebra Λ simplify drastically at the stage of the second syzygy; more precisely, we show that the kernel of any homomorphism between two projective left Λ-modules is isomorphic to a direct sum of principal left ideals generated by paths. As consequences, we obtain:

Finite lattices and Gröbner bases

Gröbner bases of binomial ideals arising from finite lattices will be studied. In terms of Gröbner bases and initial ideals, a characterization of finite distributive lattices as well as planar distributive lattices will be given.     

Syzygies among reduction operators

We introduce the notion of syzygy for a set of reduction operators and relate it to the notion of syzygy for presentations of algebras. We give a method for constructing a linear basis of the space of syzygies for a set of reduction operators. We interpret these syzygies in terms of the confluence property from rewriting theory. This enables us to optimise the completion procedure for reduction operators based on a criterion for detecting useless reductions. We illustrate this criterion with an example of construction of commutative Gröbner basis.     

Über den arithmetischen Rang quadratfreier Potenzproduktideale

In this paper we give an algorithm to compute an upper bound for the arithmetical rank of squarefree monomial ideals, i.e. the minimal number of hypersurfaces which cut out set-theoretically the variety of such an ideal. An apriori bound N – a=b+2 is obtained, where N means the number of variables, a the lowest degree in the ideal and b the lowest degree of syzygies in the first syzygy module (Thm. 2). These results sharpen more general results of [2] for the considered class of ideals by methods different from [1], [7].     

多项式组的特征分解

任一多项式理想的特征对是指由该理想的约化字典序Grobner基G和含于其中的极小三角列C构成的有序对(G,C).当C为正则列或正规列时,分别称特征对(G,C)为正则的或正规的.当G生成的理想与C的饱和理想相同时,称特征对(G,C)为强的.一组多项式的(强)正则或(强)正规特征分解是指将该多项式组分解为有限多个(强)正则或(强)正规特征对,使其满足特定的零点与理想关系.本文简要回顾各种三角分解及相应零点与理想分解的理论和方法,然后重点介绍(强)正则与(强)正规特征对和特征分解的性质,说明三角列、Ritt特征列和字典序Grobner基之间的内在关联,建立特征对的正则化定理以及正则、正规特征对的强化方法,进而给出两种基于字典序Grobner基计算、按伪整除关系分裂和构建、商除可除理想等策略的(强)正规与(强)正则特征分解算法.这两种算法计算所得的强正规与强正则特征对和特征分解都具有良好的性质,且能为输入多元多项式组的零点提供两种不同的表示.本文还给出示例和部分实验结果,用以说明特征分解方法及其实用性和有效性.     

On constructing bases for ideals in polynomial rings over the integers

The author defines canonical bases for ideals in polynomial rings over Z and develops an algorithm for constructing such a basis for a given ideal. Results of previous authors are discussed and a comparison between those and the results obtained here is included.     

Parallel algorithms for Gröbner-basis construction

The problem of the Gröbner-basis construction is important both from the theoretical and applied points of view. As examples of applications of Gröbner bases, one can mention the consistency problem for systems of nonlinear algebraic equations and the determination of the number of solutions to a system of nonlinear algebraic equations. The Gröbner bases are actively used in the constructive theory of polynomial ideals and at the preliminary stage of numerical solution of systems of nonlinear algebraic equations. Unfortunately, many real examples cannot be processed due to the high computational complexity of known algorithms for computing the Gröbner bases. However, the efficiency of the standard basis construction can be significantly increased in practice. In this paper, we analyze the known algorithms for constructing the standard bases and consider some methods for increasing their efficiency. We describe a technique for estimating the efficiency of paralleling the algorithms and present some estimates.     

GROEBNER BASES FOR CONSTRUCTIBLE SETS

ABSTRACT

A constructible set can be defined in terms of a unique sequence of varieties. Given a monomial ordering, the reduced Groebner bases of the ideals of these varieties comprise a complete invariant for the constructible set. Morphic images of varieties, such as orbits under algebraic group actions and projections of varieties, can therefore be assigned complete invariants.     

Gröbner Bases for Ideals of σ-PBW Extensions

In this article, we introduce the σ-PWB extensions and construct the theory of Gröbner bases for the left ideals of them. We prove the Hilbert's basis theorem and the division algorithm for this more general class of Poincaré–Birkhoff–Witt extensions. For the particular case of bijective and quasi-commutative σ-PWB extensions, we implement the Buchberger's algorithm for computing Gröbner bases of left ideals.     

The free product of algebras with standard base for ideals

A wide class of associative algebras admitting standard bases of ideals was introduced in previous papers of the author. They are referred to as algebras with strong filtration. This class includes all known examples of associative algebras whose ideals possess standard bases. It was proved by the author that the class of algebras with strong filtration is closed with respect to direct sums and tensorial products. In this paper it is shown that this class is also closed with respect to free products of algebras.     

Supernormal Vector Configurations

A configuration of lattice vectors is supernormal if it contains a Hilbert basis for every pointed cone spanned by a subset. We study such configurations from various perspectives, including triangulations, integer programming and Gröbner bases. Our main result is a bijection between virtual chambers of the configuration and virtual initial ideals of the associated binomial ideal.     

Generalized binomial edge ideals

This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Gröbner basis can be computed by studying paths in the graph. Since these Gröbner bases are square-free, generalized binomial edge ideals are radical. To find the primary decomposition a combinatorial problem involving the connected components of subgraphs has to be solved. The irreducible components of the solution variety are all rational.     

The Lefschetz Property for Componentwise Linear Ideals and Gotzmann Ideals

Abstract

For standard graded Artinian K-algebras defined by componentwise linear ideals and Gotzmann ideals, we give conditions for the weak Lefschetz property in terms of numerical invariants of the defining ideals.