179 research outputs found

    The Slice Algorithm For Irreducible Decomposition of Monomial Ideals

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    Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial ideals. The paper includes experiments showing good performance in practice.Comment: 25 pages, 8 figures. See http://www.broune.com/ for the data use

    Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals

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    An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by finding the space of dual functionals that annihilate it, reducing the problem to one of linear algebra. There are several known algorithms for finding the truncated dual up to any specified degree, which is useful for describing zero-dimensional ideals. We present a stopping criterion for positive-dimensional cases based on homogenization that guarantees all generators of the initial monomial ideal are found. This has applications for calculating Hilbert functions.Comment: 19 pages, 4 figure

    Primary Components of Binomial Ideals

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    Binomials are polynomials with at most two terms. A binomial ideal is an ideal generated by binomials. Primary components and associated primes of a binomial ideal are still binomial over algebraically closed fields. Primary components of general binomial ideals over algebraically closed fields with characteristic zero can be described combinatorially by translating the operations on binomial ideals to operations on exponent vectors. In this dissertation, we obtain more explicit descriptions for primary components of special binomial ideals. A feature of this work is that our results are independent of the characteristic of the field. First of all, we analyze the primary decomposition of a special class of binomial ideals, lattice ideals, in which every variable is a nonzerodivisor modulo the ideal. Then we provide a description for primary decomposition of lattice ideals in fields with positive characteristic. In addition, we study the codimension two lattice basis ideals and we compute their primary components explicitly. An ideal I ⊆ k[x_(1),….x_(n) ] is cellular if every variable is either a nonzerodivisor modulo I or is nilpotent modulo I. We characterize the minimal primary components of cellular binomial ideals explicitly. Another significant result is a computation of the Hull of a cellular binomial ideal, that is the intersection of all of its minimal primary components. Lastly, we focus on commutative monoids and their congruences. We study properties of monoids that have counterparts in the study of binomial ideals. We provide a characterization of primary ideals in positive characteristic, in terms of the congruences they induce

    Generating Polynomials and Symmetric Tensor Decompositions

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    This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial. The homogenization of a generating polynomial belongs to the apolar ideal of the tensor. A symmetric tensor decomposition can be determined by a set of generating polynomials, which can be represented by a matrix. We call it a generating matrix. Generally, a symmetric tensor decomposition can be determined by a generating matrix satisfying certain conditions. We characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness). Using these properties, we propose methods for computing symmetric tensor decompositions. Extensive examples are shown to demonstrate the efficiency of proposed methods.Comment: 35 page

    Siphons in chemical reaction networks

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    Siphons in a chemical reaction system are subsets of the species that have the potential of being absent in a steady state. We present a characterization of minimal siphons in terms of primary decomposition of binomial ideals, we explore the underlying geometry, and we demonstrate the effective computation of siphons using computer algebra software. This leads to a new method for determining whether given initial concentrations allow for various boundary steady states

    Polyhedral Cones of Magic Cubes and Squares

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    Using computational algebraic geometry techniques and Hilbert bases of polyhedral cones we derive explicit formulas and generating functions for the number of magic squares and magic cubes.Comment: 14 page
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