scholarly journals Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

2021 ◽  
Author(s):  
Manuel Baptista Branco ◽  
Isabel Colaço ◽  
Ignacio Ojeda
Keyword(s):  
Minimal System ◽  
Toric Ideal ◽  
Monomial Curves ◽  
Minimal System Of Generators ◽  
Positive Integers ◽  
The Ideal

Let $a, b$ and $n > 1$ be three positive integers such that $a$ and $\sum_{j=0}^{n-1} b^j$ are relatively prime. In this paper, we prove that the toric ideal $I$ associated to the submonoid of $\mathbb{N}$ generated by $\{\sum_{j=0}^{n-1} b^j\} \cup \{\sum_{j=0}^{n-1} b^j + a\, \sum_{j=0}^{i-2} b^j \mid i = 2, \ldots, n\}$ is determinantal. Moreover, we prove that for $n > 3$, the ideal $I$ has a unique minimal system of generators if and only if $a < b-1$.

scholarly journals Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3204
Author(s):  
Manuel B. Branco ◽  
Isabel Colaço ◽  
Ignacio Ojeda
Keyword(s):  
Minimal System ◽  
Toric Ideal ◽  
Monomial Curves ◽  
Minimal System Of Generators ◽  
Positive Integers ◽  
The Ideal

Let a,b and n>1 be three positive integers such that a and ∑j=0n−1bj are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of N generated by {∑j=0n−1bj}∪{∑j=0n−1bj+a∑j=0i−2bj∣i=2,…,n} is determinantal. Moreover, we prove that for n>3, the ideal I has a unique minimal system of generators if and only if a<b−1.

Implosive graphs: Square-free monomials on symbolic Rees algebras

2016 ◽  
Vol 16 (08) ◽  
pp. 1750145 ◽  
Author(s):  
A. Flores-Méndez ◽  
I. Gitler ◽  
E. Reyes
Keyword(s):  
Structural Properties ◽  
Monomial Ideal ◽  
Minimal System ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Vertex Set ◽  
Minimal System Of Generators

Let [Formula: see text] be the edge monomial ideal of a graph [Formula: see text], whose vertex set is [Formula: see text]. [Formula: see text] is implosive if the symbolic Rees algebra [Formula: see text] of [Formula: see text] has a minimal system of generators [Formula: see text] where [Formula: see text] are square-free monomials. We give some structural properties of implosive graphs and we prove that they are closed under clique-sums and odd subdivisions. Furthermore, we prove that universally signable graphs are implosive. We show that odd holes, odd antiholes and some Truemper configurations (prisms, thetas and even wheels) are implosive. Moreover, we study excluded families of subgraphs for the class of implosive graphs. In particular, we characterize which Truemper configurations and extensions of odd holes and antiholes are minimal nonimplosive.

A multiplicative property of R-sequences and H1-sets

1975 ◽  
Vol 78 (1) ◽  
pp. 1-6 ◽  
Author(s):  
C. P. L. Rhodes
Keyword(s):  
Commutative Ring ◽  
Multiplicative Property ◽  
Rees Algebras ◽  
Image Position ◽  
Positive Integers ◽  
The Ideal

Let R be a commutative ring which may not contain a multiplicative identity. A set of elements a1,…,ak in R will be called an H1-set (this notation is explained in section 1) if for each relation r1a1 + … +rkak = 0 (ri ∈ R) there exist elements sij ∈ R such thatwhere Xl,…,Xk are indeterminates. Any R-sequence is an H1-set, but there do exist H1-sets which are not R-sequences (see section 1). Throughout this note we consider an H1-set a1,…,ak which we suppose to be partitioned into two non-empty sets bl…, br and cl,…, cs. Our main purpose is to show that the ideals B = Rb1 + … + Rbr and C = Rc1 + … + Rcs satisfy Bm ∩ Cn = BmCn for all positive integers m and n (Corollary 1). This generalizes Lemma 2 of Caruth(2) where the result is proved when a1,…, ak is a permutable R-sequence. Our proof involves more detail than is necessary just for this, and we obtain various other properties of H1-sets. In particular we extend the main results of Corsini(3) concerning the symmetric and Rees algebras of a power of the ideal Ra1 +… + Rak (Corollary 3).

scholarly journals Schubert presentation of the cohomology ring of flag manifolds

2015 ◽  
Vol 18 (1) ◽  
pp. 489-506 ◽  
Author(s):  
Haibao Duan ◽  
Xuezhi Zhao
Keyword(s):  
Lie Group ◽  
Maximal Torus ◽  
Cohomology Ring ◽  
Minimal System ◽  
Schubert Calculus ◽  
Flag Manifolds ◽  
Generators And Relations ◽  
Integral Cohomology ◽  
Minimal System Of Generators ◽  
Connected Lie Group

Let $G$ be a compact connected Lie group with a maximal torus $T$. In the context of Schubert calculus we present the integral cohomology $H^{\ast }(G/T)$ by a minimal system of generators and relations.

scholarly journals Ideal convergence in locally solid Riesz spaces

Filomat ◽  
2014 ◽  
Vol 28 (4) ◽  
pp. 797-809 ◽  
Author(s):  
Bipan Hazarika
Keyword(s):  
Bounded Sequence ◽  
Riesz Space ◽  
Ideal Convergence ◽  
Riesz Spaces ◽  
Basic Properties ◽  
Locally Solid Riesz Space ◽  
Positive Integers ◽  
The Ideal

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. In this paper, we introduce the concepts of ideal ?-convergence, ideal ?-Cauchy and ideal ?-bounded sequence in locally solid Riesz space endowed with the topology ?. Some basic properties of these concepts has been investigated. We also examine the ideal ?-continuity of a mapping defined on locally solid Riesz space.

scholarly journals Ideal solutions of the Tarry-Escott problem of degree eleven with applications to sums of thirteenth powers.

2008 ◽  
Vol Volume 31 ◽  
Author(s):  
Ajai Choudhry ◽  
Jaroslaw Wroblewski
Keyword(s):  
Diophantine Equations ◽  
Numerical Solutions ◽  
Infinite Family ◽  
Simultaneous Equations ◽  
Simultaneous Diophantine Equations ◽  
Ideal Solutions ◽  
International Audience ◽  
Positive Integers ◽  
The Ideal

International audience This paper is concerned with the system of simultaneous diophantine equations $\sum_{i=1}^6A_i^k=\sum_{i=1}^6B_i^k$ for $k=2, 4, 6, 8, 10.$ Till now only two numerical solutions of the system are known. This paper provides an infinite family of solutions. It is well-known that solutions of the above system lead to ideal solutions of the Tarry-Escott Problem of degree $11$, that is, of the system of simultaneous equations, $\sum_{i=1}^{12}a_i^k=\sum_{i=1}^{12}b_i^k$ for $k=1, 2, 3,\ldots,11.$ We use one of the ideal solutions to prove new results on sums of $13^{th}$ powers. In particular, we prove that every integer can be expressed as a sum or difference of at most $27$ thirteenth powers of positive integers.

scholarly journals On the structure of local cohomology modules for monomial curves in

1994 ◽  
Vol 136 ◽  
pp. 81-114 ◽  
Author(s):  
H. Bresinsky ◽  
F. Curtis ◽  
M. Fiorentini ◽  
L. T. Hoa
Keyword(s):  
Local Cohomology ◽  
Closed Field ◽  
Ideal Sheaf ◽  
Homogeneous Ideal ◽  
Local Cohomology Module ◽  
Algebraic Set ◽  
Monomial Curves ◽  
Local Cohomology Modules ◽  
Cohomology Modules ◽  
The Ideal

Our setting for this paper is projective 3-space over an algebraically closed field K. By a curve C ⊂ is meant a 1-dimensional, equidimensional projective algebraic set, which is locally Cohen-Macaulay. Let be the Hartshorne-Rao module of finite length (cf. [R]). Here Z is the set of integers and ℐc the ideal sheaf of C. In [GMV] it is shown that , where is the homogeneous ideal of C, is the first local cohomology module of the R-module M with respect to . Thus there exists a smallest nonnegative integer k ∊ N such that , (see also the discussion on the 1-st local cohomology module in [GW]). Also in [GMV] it is shown that k = 0 if and only if C is arithmetically Cohen-Macaulay and C is arithmetically Buchsbaum if and only if k ≤ 1. We therefore have the following natural definition.

scholarly journals Some Families of Componentwise Linear Monomial Ideals

2007 ◽  
Vol 187 ◽  
pp. 115-156 ◽  
Author(s):  
Christopher A. Francisco ◽  
Adam Van Tuyl
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Monomial Ideals ◽  
Fat Points ◽  
Graded Betti Numbers ◽  
Alexander Duality ◽  
Componentwise Linear Ideals ◽  
Positive Integers ◽  
The Ideal

AbstractLet R = k[x1,…,xn] be a polynomial ring over a field k. Let J = {j1,…,jt} be a subset of {1,…, n}, and let mJ ⊂ R denote the ideal (xj1,…,xjt). Given subsets J1,…,Js of {1,…, n} and positive integers a1,…,as, we study ideals of the form These ideals arise naturally, for example, in the study of fat points, tetrahedral curves, and Alexander duality of squarefree monomial ideals. Our main focus is determining when ideals of this form are componentwise linear. Using polymatroidality, we prove that I is always componentwise linear when s ≤ 3 or when Ji ∪ Jj = [n] for all i ≠ j. When s ≥ 4, we give examples to show that I may or may not be componentwise linear. We apply these results to ideals of small sets of general fat points in multiprojective space, and we extend work of Fatabbi, Lorenzini, Valla, and the first author by computing the graded Betti numbers in the s = 2 case. Since componentwise linear ideals satisfy the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan when char(k) = 0, our work also yields new cases in which this conjecture holds.

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