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Abstract: This paper describes an algorithm which produces all ideals compatible with a given surjective Frobenius near-splitting. [Copyright &y& Elsevier]

Abstract: This paper introduces some relations about Cayley graphs and -graphs. We present a sufficient condition to recognize when a -graph is a Cayley graph. The relation between -graphs and Cayley graphs allows us to consider some applications to the hamiltonicity of Cayley graphs. In the last section we illustrate our results by showing that some new classes of Cayley graphs are hamiltonian. [Copyright &y& Elsevier]

Abstract: Every coherent system has a monomial ideal associated with it and the knowledge of its multigraded Betti numbers provides reliability bounds for the corresponding system, which are the tightest among a certain class of such bounds. Some alternative methods for computing the multigraded Betti numbers are used in this paper and applied in the study of reliability. We obtain special results for well known examples and show that computational commutative algebra techniques can be used beneficially in the reliability analysis of systems of different types. [Copyright &y& Elsevier]

Abstract: A connected algebraic group in characteristic is uniquely determined by its Lie algebra. In this paper algorithms are given for constructing an algebraic group in characteristic , given its Lie algebra. In particular, this leads to an algorithm for finding a maximal reductive subgroup and the unipotent radical of an algebraic group. [Copyright &y& Elsevier]

Abstract: This paper presents a new algorithm that computes the local algebras of the roots of a zero-dimensional polynomial equation system, with a number of operations in the coefficient field that is polynomial in the number of variables, in the evaluation cost of the equations and in a Bézout number. [Copyright &y& Elsevier]

Abstract: The classification of naturally graded quasi-filiform Lie algebras is known; they have the characteristic sequence where is the dimension of the algebra. In the present paper we deal with naturally graded quasi-filiform non-Lie–Leibniz algebras which are described by the characteristic sequence or . The first case has been studied in [Camacho, L.M., Gómez, J.R., González, A.J., Omirov, B.A., 2006. Naturally graded 2-filiform Leibniz Algebra and its applications, preprint, MA1-04-XI06] and now, we complete the classification of naturally graded quasi-filiform Leibniz algebras. For this purpose we use the software (the program used is explained in the last section). [Copyright &y& Elsevier]

Abstract: Lawrence semigroups arise as a tool to compute Graver bases of semigroup ideals. It is known that the minimal free resolution of semigroup ideals is characterized by the reduced homologies of certain simplicial complexes. In this paper we study the minimal degrees of a Lawrence semigroup ideal and its first syzygy given a combinatorial characterization of the nonvanishing cycles in their associated reduced homologies. We specialize the results that appeared in [Briales, E., Campillo, A., Marijuán, C., Pisón, P., 1998. Minimal systems of generators for ideals of semigroups. J. Pure Appl. Algebra, 127, 7–30] and [Pisón-Casares, P., Vigneron-Tenorio, A., 2001. First syzygies of toric varieties and diophantine equations in congruence. Comm. Alg. 29 (4), 1445–1466] to the Lawrence semigroups. [Copyright &y& Elsevier]

Abstract: Consider the class of exp–log constants, which is constructed from the integers using the field operations, exponentiation and logarithm. Let be such an exp–log constant and let be its size as an expression. Witness conjectures attempt to give bounds for the number of decimal digits which need to be evaluated in order to test whether equals zero. For this purpose, it is convenient to assume that exponentials are only applied to arguments with absolute values bounded by 1. In that context, several witness conjectures have appeared in the literature and the strongest one states that it is possible to choose . In this paper we give a counterexample to this conjecture. We also extend it so as to cover similar, polynomial witness conjectures. [Copyright &y& Elsevier]

Abstract: Let be a field and a finite directed multi-graph. The focus of this paper is to offer a complete description of all path algebras and admissible orders with the property that all of their finitely generated ideals have finite Gröbner bases and of those which contain a finitely generated ideal whose Gröbner bases are all infinite. [Copyright &y& Elsevier]

Abstract: Algebraic function fields of positive characteristic are non-perfect fields, and many standard algorithms for solving some fundamental problems in commutative algebra simply do not work over these fields. This paper presents practical algorithms for the first time for (1) computing the primary decomposition of ideals of polynomial rings defined over such fields and (2) factoring arbitrary multivariate polynomials over such fields. Difficulties involving inseparability and the situation where the transcendence degree is greater than one are completely overcome, while the algorithms avoid explicit construction of any extension of the input base field. As a corollary, the problem of computing the primary decomposition of a positive-dimensional ideal over a finite field is also solved. The algorithms perform very effectively in an implementation within the Magma Computer Algebra System, and an analysis of their practical performance is given. [Copyright &y& Elsevier]

Abstract: In this paper, we present different results related to bezoutian and residue theory. We consider, in particular, the problem of computing the structure of the quotient ring by an affine complete intersection, and an algorithm to obtain it, as conjectured in [Cardinal, J.-P., 1993. Dualité et algorithmes itératifs pour la résolution de systèmes polynomiaux. Ph.D. Thesis, Univ. de Rennes]. We analyze it in detail and prove the validity of the conjecture, for a modification of the initial method. Direct applications of the results in effective algebraic geometry are given. [Copyright &y& Elsevier]

The purpose of this paper is to present an effective method of deciding the semidefiniteness of multivariate polynomials with coefficients in a computable ordered field, which admits an effective method of finding an isolating set for every non-zero univariate polynomial. Based on this method, the decision of the semidefiniteness of a multivariate polynomial may be reduced to testing some resulted polynomials in fewer variables, of which the total degrees and the term numbers do not exceed those of the given polynomial. With the aid of the computer algebra system Maple, our method is used to solve several examples. [Copyright &y& Elsevier]

We correct four typographical errors in the paper, “The F5 criterion revised.” [ABSTRACT FROM AUTHOR]