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Computing lattice ideals of unions of monomial curves

Marı́a-Jesús Pisabarro
2004 Journal of symbolic computation  
We find an algorithm that checks whether or not the ideal of a union of monomial curves is binomial and another one that calculates curves such that their associated ideal is a prescribed lattice ideal  ...  In this paper we present a combinatorial study of binomial ideals of dimension 1 of k[X 1 , . . . , X n ], using monomial parametrizations of the irreducible affine curves defined by their associated primes  ...  In particular, in Proposition 4 we prove that the ideal of a monomial curve is the sum of a prime lattice ideal and a monomial ideal, and we compute the lattice and the partial character starting from  ... 
doi:10.1016/j.jsc.2004.03.004 fatcat:edevn4lurvbgxptvat4t3xrrsy

Contents of Volumes 37 and 38

2004 Journal of symbolic computation  
-J., Computing lattice ideals of unions of monomial curves . . 1025 Sills, A.V., RRtools-a Maple package for aiding the discovery and proof of finite Rogers-Ramanujan type identities .. . . . . . .  ...  ., Computing minimal generators of the ideal of a general projective curve . . . . . . . . . . . . . . . . 295 Burckel, S., Elementary decompositions of arbitrary maps over finite sets . . 305 Famelis  ... 
doi:10.1016/s0747-7171(04)00108-7 fatcat:tmvrkoolp5b4jkrpklob3tet5q

Index to Volumes 37 and 38

2004 Journal of symbolic computation  
of monomial algebras, 537 Computing maximal subgroups of finite groups, 589 Computing minimal generators of the ideal of a general projective curve, 295 Corner edge cutting and Dixon A-resultant quotients  ...  ideal of a general projective curve, 295 BELABAS, K., A relative van Hoeij algorithm over number fields, 641 BERNSTEIN, D., The computational complexity of rules for the character table of S n , 727 BURCKEL  ...  -J., Computing lattice ideals of unions of monomial curves, 1025 Polynomial factorization algorithms over number fields, 1429 Resultants for unmixed bivariate polynomial systems produced using the Dixon  ... 
doi:10.1016/s0747-7171(04)00109-9 fatcat:q3cckydpknhjhinygacsvlj52y

Short rational functions for toric algebra and applications

J.A. De Loera, D. Haws, R. Hemmecke, P. Huggins, B. Sturmfels, R. Yoshida
2004 Journal of symbolic computation  
Under the assumption that d and n are fixed, this representation allows us to compute a universal Gröbner basis and the reduced Gröbner basis of the ideal I A , with respect to any term order, in time  ...  We describe other applications, such as the computation of Hilbert series of normal semigroup rings, and we indicate applications to enumerative combinatorics, integer programming, and statistics.  ...  The row sums and column sums of the entries are given there too. Using LattE we obtained the exact answer 8813835312287964978894.  ... 
doi:10.1016/j.jsc.2004.02.001 fatcat:lbwon7ivrfh3bclq4g34cfbacu

Stanley-Reisner rings and the radicals of lattice ideals [article]

Anargyros Katsabekis, Marcel Morales, Apostolos Thoma
2003 arXiv   pre-print
In this article we associate to every lattice ideal I_L,ρ⊂ K[x_1,..., x_m] a cone σ and a graph G_σ with vertices the minimal generators of the Stanley-Reisner ideal of σ .  ...  This result provides a lower bound for the minimal number of generators of I_L,ρ and therefore improves the generalized Krull's principal ideal theorem for lattice ideals.  ...  In section 5 we compute these bounds for a special class of lattice ideals.  ... 
arXiv:math/0310313v1 fatcat:nqluvi4yibgbfhztywp5sd45x4

Page 984 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
ideals of lattice ideals.  ...  Let LC Z”" be a lattice. The lattice ideal J; is the ideal of K[x,---,Xn] generated by the binomials x“ — x? with a—be L.  ... 

Syzygies of codimension 2 lattice ideals

Irena Peeva, Bernd Sturmfels
1998 Mathematische Zeitschrift  
The key ingredient is that no monomial is a zero-divisor modulo I L , and this property of toric ideals holds for lattice ideals as well. Corollary 2.2.  ...  The congruence class C is a translate of the r-dimensional lattice L. Hence C is an affine lattice of rank r.  ...  For codimension 2 lattice ideals it is advantageous to compute Gröbner bases by calling upon a subroutine for computing minimal generators. Algorithm 8.2.  ... 
doi:10.1007/pl00004645 fatcat:nhzvhamke5fczlgwlxamw6r7l4

Tropical secant graphs of monomial curves

María Angélica Cueto, Shaowei Lin
2010 Discrete Mathematics & Theoretical Computer Science  
Using techniques from tropical geometry, we give algorithms to effectively compute this degree (as well as its multidegree) and the Newton polytope of the first secant variety of any given monomial curve  ...  Using this graph, we construct the tropicalization of the first secant variety of a monomial projective curve with exponent vector $(0, i_1, \ldots, i_n)$, which can be described by a balanced graph called  ...  Combinatorics of Monomial Curves In this section, we compute the tropical variety of the surface Z described in Theorem 6.  ... 
doi:10.46298/dmtcs.2820 fatcat:46xaml4wlbc6jb6yz5hwbfvw6e

Stanley–Reisner rings and the radicals of lattice ideals

Anargyros Katsabekis, Marcel Morales, Apostolos Thoma
2006 Journal of Pure and Applied Algebra  
This result provides a lower bound for the minimal number of generators of I L, which improves the generalized Krull's principal ideal theorem for lattice ideals.  ...  In this article we associate to every lattice ideal I L, ⊂ K[x 1 , . . . , x m ] a cone and a simplicial complex with vertices the minimal generators of the Stanley-Reisner ideal of .  ...  In Section 6 we compute these bounds for a special class of lattice ideals.  ... 
doi:10.1016/j.jpaa.2005.06.005 fatcat:zafofpowijeyxkixdmamd6b5zi

Forty questions on singularities of algebraic varieties

Herwig Hauser, Josef Schicho
2011 Asian Journal of Mathematics  
For curves, computing of adjoints is basically equivalent to computing the normalization.  ...  Clearly, any union of linear spaces is mikado, as well as any union of smooth plane curves meeting pairwise transversally.  ...  (The Weierstrass division is defined by requiring that no monomial summand of the remainder is divisible by the leading monomial of f (0, . . . , 0, x n .)  ... 
doi:10.4310/ajm.2011.v15.n3.a5 fatcat:67c7v3uejvflpbk7n5eh5nfjhu

Multiplier ideals of monomial space curves [article]

Howard M Thompson
2011 arXiv   pre-print
The goal of this paper is to produce a formula for the multiplier ideals of monomial space curves in the spirit of Howald's formula for the multiplier ideals of monomial ideals.  ...  This is achieved by constructing a toric blowup of affine space in such a way that a log resolution of the monomial curve may be constructed from this toric variety in a well controlled manner.  ...  This theorem tells us that it suffices to consider the divisors that appear on the blowup of the monomial ideal and it gives us a convenient way to compute the intersection (1) in the monomial case.  ... 
arXiv:1006.1915v4 fatcat:gyx3yim33zdanmcmgay3culoke

Theory and Applications of Lattice Point Methods for Binomial Ideals [chapter]

Ezra Miller
2011 Combinatorial Aspects of Commutative Algebra and Algebraic Geometry  
This survey of methods surrounding lattice point methods for binomial ideals begins with a leisurely treatment of the geometric combinatorics of binomial primary decomposition.  ...  Lattice-point combinatorics related to monoids and congruences has recently been shown relevant to the theory of combinatorial games, and is sure to play a key role in algorithms for computing rational  ...  Monomial Ideals and Primary Binomial Ideals The lattice-point geometry of binomial primary decomposition generalizes the geometry of monomial ideals.  ... 
doi:10.1007/978-3-642-19492-4_8 fatcat:yrwpp4nwgnf7rbwjntf6wkvyz4

Standard pairs and group relaxations in integer programming

Serkan Hoşten, Rekha R. Thomas
1999 Journal of Pure and Applied Algebra  
of initial ideals of monomial curves.  ...  The tools used are those of standard pair decompositions of standard monomials of a toric initial ideal, localizations of such ideals at their associated primes and group relaxations of integer programs  ...  Neither monomial is divisible by x i x j . Theorem 5 . 4 . 54 The ideal inc (I L ) is contained in M = j ⊃ M j . Moreover; the standard pairs of M is the union of the standard pairs of M j .  ... 
doi:10.1016/s0022-4049(99)00009-2 fatcat:qlvryn4ywnglzcqc63szmrqhfi

Coordinate rings for the moduli stack of quasi-parabolic principal bundles on a curve and toric fiber products

Christopher Manon
2012 Journal of Algebra  
on a generic marked projective curve.  ...  We show that many of results on the projective coordinate rings of M_C, p⃗(SL_2()) follow from closure properties of this category with respect to fiber products.  ...  the behavior of graded algebras in flat families.  ... 
doi:10.1016/j.jalgebra.2012.05.007 fatcat:6oyaav2hmjhulgg57ooim326yy

Degenerations of Monomial Ideals

Heather Russell
2004 Mathematical Research Letters  
We describe the degenerations of monomial ideals in K [[x, y]] with Aut(K[[x, y]])-orbit of dimension at most 3.  ...  In particular, we determine the monomial ideals that any power of (x, y 4 ) can degenerate to and make a conjecture about all the ideals that the powers of (x, y 4 ) can degenerate to.  ...  The intersections of these curves correspond to monomial ideals, since these are the T -invariant ideals.  ... 
doi:10.4310/mrl.2004.v11.n2.a7 fatcat:zh5s3kv7cnfmrnx6ctwwfiv7vm
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