有限域上多项式集的简单分解

研究如何将任意有限域上的多项式集分解为有限多个简单列.为了解决这一问题,首先研究简单列和根理想之间的关系,然后基于已有的正则分解算法和有限域上理想的根的两种计算方法设计一个有限域上多项式集的简单分解算法.计算试验表明,文章给出的算法是有效的.     

Binomial Canonical Decompositions of Binomial Ideals

In this article, we prove that every binomial ideal in a polynomial ring over an algebraically closed field of characteristic zero admits a canonical primary decomposition into binomial ideals. Moreover, we prove that this special decomposition is obtained from a cellular decomposition which is also defined in a canonical way and does not depend on the field.     

Reverse-engineering of polynomial dynamical systems

Multivariate polynomial dynamical systems over finite fields have been studied in several contexts, including engineering and mathematical biology. An important problem is to construct models of such systems from a partial specification of dynamic properties, e.g., from a collection of state transition measurements. Here, we consider static models, which are directed graphs that represent the causal relationships between system variables, so-called wiring diagrams. This paper contains an algorithm which computes all possible minimal wiring diagrams for a given set of state transition measurements. The paper also contains several statistical measures for model selection. The algorithm uses primary decomposition of monomial ideals as the principal tool. An application to the reverse-engineering of a gene regulatory network is included. The algorithm and the statistical measures are implemented in Macaulay 2, and are available from the authors.     

ALGORITHMS ON PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS

ABSTRACT

Heinzer, Mirbagheri, Ratliff, and Shah investigate parametric decomposition of monomial ideals on a regular sequence of a commutative ring R with identity 1 and prove that if every finite intersection of monomial ideals in R is again a monomial ideal, then each monomial ideal has a unique irredundant parametric decomposition. Sturmfels, Trung, and Vogels prove a similar result without the uniqueness. Bayer, Peeva, and Strumfels study generic monomial ideals, that is monomial ideals in the polynomial ring such that no variable appears with the same nonzero exponent in two different minimal generators, and for these ideals they prove the uniqueness of the irredundant irreducible decompositions and give an algorithm to construct this unique irredundant irreducible decomposition. In this paper, we present three algorithms for finding the unique irredundant irreducible decomposition of any monomial ideal.     

Genus number and l-rank of genus group of cyclic extensions of degree l

Let l be an odd prime, and k an algebraic number field of a finite degree. Let S be a finite product of distinct prime ideals g of k such that Ng1 (mod l). Let I(s) (resp. P(s)) denote the group of ideals (resp. principal ideals) of k prime to S, and let P_S denote the ray modulo S. In this paper we prove that the order (resp. the l-rank) of I(S)/P(S)^lP_S is expressed by the decomposition groups of prime factors of S in a Galois extension K_o (resp. K_r) over k. As an application of this, some results about genus theory are obtained.     

Primary decomposition of modules: A computational differential approach

We study primary submodules and primary decompositions from a differential and computational point of view. Our main theoretical contribution is a general structure theory and a representation theorem for primary submodules of an arbitrary finitely generated module over a polynomial ring. We characterize primary submodules in terms of differential operators and punctual Quot schemes. Moreover, we introduce and implement an algorithm that computes a minimal differential primary decomposition for a module.     

Reflexive Property of Rings

Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this article, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings.     

On the factorization of polynomial matrices over the domain of principal ideals

We consider the problem of decomposition of polynomial matrices over the domain of principal ideals into a product of factors of lower degrees with given characteristic polynomials. We establish necessary and, under certain restrictions, sufficient conditions for the existence of the required factorization.     

Gröbner Bases for the Polynomial Ring with Infinite Variables and Their Applications

We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-to-one correspondences among various sets of partitions by using the division algorithm.     

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.     

Über B. Buchbergers verfahren,systeme algebraischer gleichungen zu lösen

An algorithm of B. Buchberger's is extended to polynomial rings over a Noetherian ring. In a specialized version, it can be used for computing “elimination ideals”. Over fields, it provides the determination of the minimal prime ideals which contain the given ideal, except that the primeness must be proved with other methods. Estimates for computing time are not given.     

Peterson–Gorenstein–Zierler algorithm for skew RS codes

We design a non-commutative version of the Peterson–Gorenstein–Zierler decoding algorithm for a class of codes that we call skew RS codes. These codes are left ideals of a quotient of a skew polynomial ring, which endow them of a sort of non-commutative cyclic structure. Since we work over an arbitrary field, our techniques may be applied both to linear block codes and convolutional codes. In particular, our decoding algorithm applies for block codes beyond the classical cyclic case.     

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.     

有限域上理想的准素分解

主要研究有限域上零维理想的准素分解问题,时文[1]的所提到的方法和理想商环不变子空阃的基元素的可分性进行讨论,并给出了判定的充分必要条件。     

Optimizing a Particular Real Root of a Polynomial by a Special Cylindrical Algebraic Decomposition

We study the problem of optimizing over parameters a particular real root of a polynomial with parametric coefficients. We propose an efficient symbolic method for solving the optimization problem based on a special cylindrical algebraic decomposition algorithm, which asks for a semi-algebraic decomposition into cells in terms of number-of-roots-invariance.     

Segments and Hilbert schemes of points

Using results obtained from a study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to a degreverse term order (as, for example, the degrevlex order) in the Hilbert scheme of points in Pⁿ. In this context, we look into the properties of several types of “segment” ideals that we define and compare. This study also leads us to focus on the connections between the shape of generators of Borel ideals and the related Hilbert polynomial, thus providing an algorithm for computing all saturated Borel ideals with a given Hilbert polynomial.     

Counting solutions of polynomial systems via iterated fibrations

The paper introduces a new polynomial to count the solutions of a system of polynomial equations and inequations over an algebraically closed field of characteristic zero based on the triangular decomposition algorithm by J. M. Thomas of the nineteen-thirties. In the special case of projective varieties examples indicate that it is a finer invariant than the Hilbert polynomial. Received: 8 March 2008; Revised: 12 August 2008     

Stanley conjecture in small embedding dimension

We show that Stanley's conjecture holds for a polynomial ring over a field in four variables. In the case of polynomial ring in five variables, we prove that the monomial ideals with all associated primes of height two, are Stanley ideals.     

Growth of primary decompositions of Frobenius powers of ideals

It was previously known, by work of Smith–Swanson and of Sharp–Nossem, that the linear growth property of primary decompositions of Frobenius powers of ideals in rings of prime characteristic has strong connections to the localization problem in tight closure theory. The localization problem has recently been settled in negative by Brenner and Monsky, but the linear growth question is still open. We study growth of primary decompositions of Frobenius powers of dimension one homogeneous ideals in graded rings over fields. If the ring is positively graded we prove that the linear growth property holds. For non-negatively graded rings we are able to show that there is a “polynomial growth”. We present explicit primary decompositions of Frobenius powers of an ideal, which were known to have infinitely many associated primes, having this linear growth property. We also discuss some other interesting examples.     

Decompositions of binomial ideals

We present Binomials, a package for the computer algebra system Macaulay 2, which specializes well-known algorithms to binomial ideals. These come up frequently in algebraic statistics and commutative algebra, and it is shown that significant speedup of computations like primary decomposition is possible. While central parts of the implemented algorithms go back to a paper of Eisenbud and Sturmfels, we also discuss a new algorithm for computing the minimal primes of a binomial ideal. All decompositions make significant use of combinatorial structure found in binomial ideals, and to demonstrate the power of this approach we show how Binomials was used to compute primary decompositions of commuting birth and death ideals of Evans et al., yielding a counterexample for their conjectures.