Computing the support of monomial iterated mapping cones

In this paper we compute and manipulate the support of monomial resolutions based on iterated mapping cones. We derive in this way algorithms to obtain homological and numerical invariants of monomial ideals without actually computing their resolution. Our computations include Betti diagrams, Hilbert series and irreducible decompositions. The algorithms derived by the method presented in the paper are efficient in practice as shown by experiments.     

Value monoids of zero-dimensional valuations of rank 1

Classically, Gröbner bases are computed by first prescribing a fixed monomial order. Moss Sweedler suggested an alternative in the mid-1980s and developed a framework for performing such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on K(x,y) that are suitable for this framework. We then perform such computations for ideals in the polynomial ring K[x,y]. Interestingly, for these valuations, some ideals have finite Gröbner bases with respect to a valuation that are not Gröbner bases with respect to any monomial order, whereas other ideals only have Gröbner bases that are infinite.     

Generic and Cogeneric Monomial Ideals

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions encoded by simplicial complexes. There are numerous equivalent ways to say that a monomial ideal is generic or cogeneric. For a generic monomial ideal, the associated primes satisfy a saturated chain condition, and the Cohen–Macaulay property implies shellability for both the Scarf complex and the Stanley–Reisner complex. Reverse lexicographic initial ideals of generic lattice ideals are generic. Cohen–Macaulayness for cogeneric ideals is characterized combinatorially; in the cogeneric case, the Cohen–Macaulay type is greater than or equal to the number of irreducible components. Methods of proof include Alexander duality and Stanley’s theory of local h -vectors.     

Monomial Orderings,Rewriting Systems,and Gröbner Bases for the Commutator Ideal of a Free Algebra

In this paper we consider a free associative algebra on three generators over an arbitrary fieldK. Given a term ordering on the commutative polynomial ring on three variables overK, we construct uncountably many liftings of this term ordering to a monomial ordering on the free associative algebra. These monomial orderings are total well orderings on the set of monomials, resulting in a set of normal forms. Then we show that the commutator ideal has an infinite reduced Gröbner basis with respect to these monomial orderings, and all initial ideals are distinct. Hence, the commutator ideal has at least uncountably many distinct reduced Gröbner bases. A Gröbner basis of the commutator ideal corresponds to a complete rewriting system for the free commutative monoid on three generators; our result also shows that this monoid has at least uncountably many distinct minimal complete rewriting systems.The monomial orderings we use are not compatible with multiplication, but are sufficient to solve the ideal membership problem for a specific ideal, in this case the commutator ideal. We propose that it is fruitful to consider such, more general, monomial orderings in non-commutative Gröbner basis theory.     

The Gröbner fan and Gröbner walk for modules

This paper extends the theory of the Gröbner fan and Gröbner walk for ideals in polynomial rings to the case of submodules of free modules over a polynomial ring. The Gröbner fan for a submodule creates a correspondence between a pair consisting of a cone in the fan and a point in the support of the cone and a pair consisting of a leading monomial submodule (or equivalently, a reduced marked Gröbner basis) and a grading of the free module over the ring that is compatible with the choice of leading monomials. The Gröbner walk is an algorithm based on the Gröbner fan that converts a given Gröbner basis to a Gröbner basis with respect to a different monomial order; the point being that the Gröbner walk can be more efficient than the standard algorithms for Gröbner basis computations with difficult monomial orders. Algorithms for generating the Gröbner fan and for the Gröbner walk are given.     

On the Stability of Gröbner Bases Under Specializations

LetRbe a Noetherian commutative ring with identity,Ka field and π a ring homomorphism fromRtoK. We investigate for which ideals inR[x₁,…,x_n] and admissible orders the formation of leading monomial ideals commutes with the homomorphism π.     

An Algorithm for the Quillen–Suslin Theorem for Quotients of Polynomial Rings by Monomial Ideals

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 [ x₀, . . . ,x_n ] / I, with I a monomial ideal and k a field. Vorst proved that finitely generated projective modules over such algebras are free. Given a finitely generated module P, described by generators and relations, the algorithm tests whether P is projective, in which case it computes a free basis forP .     

An algorithm for computing compatibly Frobenius split subvarieties

This paper describes an algorithm which produces all ideals compatible with a given surjective Frobenius near-splitting.     

Taylor and Lyubeznik Resolutions via Gröbner Bases

Taylor presented an explicit resolution for arbitrary monomial ideals. Later, Lyubeznik found that a subcomplex already defines a resolution. We show that the Taylor resolution may be obtained by repeated application of the Schreyer Theorem from the theory of Gröbner bases, whereas the Lyubeznik resolution is a consequence of Buchberger’s chain criterion. Finally, we relate Fröberg’s contracting homotopy for the Taylor complex to normal forms with respect to our Gröbner bases and use it to derive a splitting homotopy that leads to the Lyubeznik complex.     

Multiple degree elevation and constrained multiple degree reduction for DP curves and surfaces

In this paper, the problem of single and multiple degree elevation and reduction for DP curves and surfaces is considered and expressed in matrix representations. Using monomial matrix representations of the univariate DP polynomials, single and multiple degree elevation matrices can be derived by minor extension of this monomial matrix. Given this result, matrices of constrained single and multiple degree reduction can be readily obtained. Finally, important and sufficient conditions for degree reduction can also be found from these resulting matrices.     

BR0代数的模糊理想与直觉模糊理想

引入BR₀代数的Fuzzy理想与素Fuzzy理想的概念;给出了Fuzzy理想与素Fuzzy理想的等价形式;在（素）Fuzzy理想基础之上又进一步引入了（素）直觉Fuzzy理想,并通过Fuzzy理想找到了它们与（素）理想之间的关系。     

BR_0代数的模糊理想与直觉模糊理想

引入BR0代数的Fuzzy理想与素Fuzzy理想的概念;给出了Fuzzy理想与素Fuzzy理想的等价形式;在(素)Fuzzy理想基础之上又进一步引入了(素)直觉Fuzzy理想,并通过Fuzzy理想找到了它们与(素)理想之间的关系。     

Generalized Ideals and Supports in Pseudo Effect Algebras

In this paper we introduce the definitions of generalized ideals and generalized filters in pseudo effect algebras. From the generalized ideals (filters), we obtain some equivalent characterization of ideals (filters). The relationships among generalized ideals, ideals and local ideals are studied in detail. We also establish an isomorphism between the set of supports and the set of generalized ideals of pseudo effect algebras.     

BCI-implicative ideals of BCI-algebras

This paper introduces the notion of BCI-implicative ideals and characterizes BCI-implicative ideals and closed BCI-implicative ideals. Using these characterizations, the connections between BCI-implicative ideals and other ideals in BCI/BCK-algebras are investigated. Additionally, the extension property of BCI-implicative ideals is established. Finally, the implicative BCI-algebras are completely described using BCI-implicative ideals. The above work generalizes the corresponding results in BCK-algebras.     

Milne’s volume function and vector symmetric polynomials

The number of real roots of a system of polynomial equations fitting inside a given box can be counted using a vector symmetric polynomial introduced by P. Milne, the volume function. We provide the expansion of Milne’s volume function in the basis of monomial vector symmetric functions, and observe that only monomial functions of a particular kind appear in the expansion, the squarefree monomial functions.     

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 Gröbner bases: every basis is determined by a certain ordering. At the same time, in order to define a Gröbner 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 Gröbner 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 Gröbner basis that completely determine this basis is presented. The paper presents also an algorithm for computing an extended universal Gröbner basis for an arbitrary zero-dimensional ideal.     

Monomial reachability and zero controllability of discrete-time positive switched systems

In this paper, monomial reachability and zero controllability properties of discrete-time positive switched systems are investigated. Necessary and sufficient conditions for these properties to hold, together with some interesting examples and some testing algorithms, are provided.     

Rough prime ideals and rough fuzzy prime ideals in semigroups

In this paper, we shall introduce the notions of rough prime ideals and rough fuzzy prime ideals in a semigroup, and give some properties of such ideals. Also, we will discuss the relations between the upper [lower] rough prime ideals and the upper [lower] approximations of their homomorphism images.     

On constructivity of involutive divisions

In the paper, the problem of existence of the minimal involutive monomial basis is considered for different involutive divisions. The existence criterion is obtained in the form of a constructivity property. In the paper, new examples of constructive and nonconstructive (> σ)-divisions are presented.     

含仿射不确定参数的多项式非线性系统控制器设计

针对含仿射时变不确定参数的多项式非线性系统，提出了基于多项式分解的控制方法. 多项式分解方法主要思想是将多项式系统转化成带自由变量的系数矩阵，从而偶次多项式的非负性验证问题可转化成线性矩阵不等式或双线性矩阵不等式求解问题. 文中多项式系统控制器综合基于 Lyapunov 稳定定理. 构造 Lyapunov 函数以及寻找反馈控制器可由所给的算法通过计算机程序自动完成. 对于多维系统相对高阶的控制器，由多项式全基构造的控制器将有很多项单项式. 为克服这一问题，文中算法给出含最少单项式的简约型控制器设计方法，并提出针对最小代价性能目标优化的增益受约次优控制. 数值仿真例子表明，文中所给的控制方法取得良好性能.