arXiv 0708.37522007-08-28math.AC

Contribution in combinatorics in commutative algebra.(ph-d thesis)

Mircea Cimpoeas

备注：: 75 pages

英文摘要：

In the first chapter we present new results related on monomial ideals of Borel type. Also, we introduce a new class of monomial ideals, called $\de$-fixed ideals, which generalize the class of $p$-Borel ideals and we extend several results to this new class. In the second chapter, we compute the generic initial ideal, with repect to the reverse lexicographic order, of an ideal which define a complete intersection of embedding dimension three with strong Lefschetz property and we show that it is an almost reverse lexicographic ideal. This enable us to give a proof for Moreno's conjecture in the case $n=3$ and characteristic zero. Also, we prove that the $d$-component of the generic initial ideal, with respect to the reverse lexicographic order, of an ideal generated by a regular sequence of homogeneous polynomials of degree $d$ is revlex, in a particular, but important, case.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Powers of Monomial Ideals and Combinatorics

This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial complexes, integral points in polytopes and graphs.

2018-09-20math.AC

Frobenius methods in combinatorics

We survey results produced from the interaction between methods in prime characteristic and combinatorial commutative algebra. We showcase results for edge ideals, toric varieties, Stanley-Reisner rings, and initial ideals that were proven via Frobenius. We also discuss results for monomial ideals obtained using Frobenius-like maps. Finally, we present results for $F$-pure rings that were inspired by work done for Stanley-Reisner rings.

2022-03-18math.AC

Euler class group of a Noetherian ring

We will study the Euler class group of a commutative Noetherian ring $A$ of dimension $d$, developed by S.M. Bhatwadekar and Raja Sridharan. This is my M.Phil thesis from 2001.

2014-08-12math.AC

Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics

We extend a few fundamental aspects of the classical theory of non-unique factorization, as presented in Geroldinger and Halter-Koch's 2006 monograph on the subject, to a non-commutative and non-cancellative setting, in the same spirit of Baeth and Smertnig's work on the factorization theory of non-commutative, but cancellative monoids [J. Algebra 441 (2015), 475-551]. Then, we bring in power monoids and, applying the abstract machinery developed in the first part, we undertake the study of their arithmetic. More in particular, let $H$ be a multiplicatively written monoid. The set $\mathcal P_…

2018-07-02math.ACmath.COmath.NT

Introduction to twisted commutative algebras

This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate rings of determinantal varieties, Segre-Veronese embeddings, and Grassmannians. The article is meant to serve as a gentle introduction to the papers of the two authors on the subject, and also to point out some literature in which these algebras appear. The first part reviews the representation theory of the symmetric groups and general linear groups. The sec…

2012-09-23math.ACmath.AGmath.RT

Invariants of ideals generated by pfaffians

Ideals generated by pfaffians are of interest in commutative algebra and algebraic geometry, as well as in combinatorics. In this article we compute multiplicity and Castelnuovo-Mumford regularity of pfaffian ideals of ladders. We give explicit formulas for some families of ideals, and indicate a procedure that allows to recursively compute the invariants of any pfaffian ideal of ladder. Our approach makes an essential use of liaison theory.

2013-03-27math.AC

更多相关论文收起更多论文

CAKR：用于基因组学的交换代数k-mer表示

CAKR: Commutative algebra k-mer representations for genomics

King Fahd University of Petroleum and Minerals（阿卜杜勒阿齐兹国王石油大学）；Michigan State University（密歇根州立大学）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；University of Georgia（佐治亚大学）；Georgia Institute of Technology（佐治亚理工学院）

本研究提出CAKR即交换代数k-mer表示框架，将多学科方法结合用于比较基因组分析，在三类任务的多数据集上优于五种现有方法，兼具可扩展性与鲁棒性。

2026-08-07math.ACq-bio.QM

锥模、二次扇与非交换解析

Conic modules, secondary fans and non-commutative resolutions

Beijing Institute of Mathematical Sciences and Applications（北京数学与应用研究所）

本文研究锥模与二次扇的关系，提出非交换解析的构造方法，并探讨了锥模在非交换解析中的应用条件。

2026-03-25math.ACmath.AG

Rees algebras, Monomial Subrings and Linear Optimization Problems

In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut pr…

2010-06-14math.AC

扫帚图与双扫帚图平方的闭邻域理想

On the closed neighborhood ideal of the square of broom and double broom graphs

Faculty of Marine Engineering, Romanian Naval Academy “Mircea cel Bătrân”（罗马尼亚海军学院“米尔恰·切·巴特兰”海事工程学院）；Faculty of Applied Sciences, National University of Science and Technology Politehnica Bucharest（布加勒斯特理工大学国家科学与技术大学应用科学学院）

本文研究无平方项单项式理想，以图等组合对象性质表达其代数不变量，聚焦扫帚图与双扫帚图平方的闭邻域理想，搭建交换代数与组合数学的关联桥梁。

2026-09-07math.ACmath.CO

Theory and applications of lattice point methods for binomial ideals

This survey of methods surrounding lattice point methods for binomial ideals begins with a leisurely treatment of the geometric combinatorics of binomial primary decomposition. It then proceeds to three independent applications whose motivations come from outside of commutative algebra: hypergeometric systems, combinatorial game theory, and chemical dynamics. The exposition is aimed at students and researchers in algebra; it includes many examples, open problems, and elementary introductions to the motivations and background from outside of algebra.

2010-09-15cs.GTmath.ACmath.CO

Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules

Let $H$ be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every $k \in \mathbb N$, let $\mathscr U_k (H)$ denote the set of all $\ell \in \mathbb N$ with the property that there are atoms $u_1, \ldots, u_k, v_1, \ldots, v_{\ell}$ such that $u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_{\ell}$ (thus, $\mathscr U_k (H)$ is the union of all sets of lengths containing $k$). The Structure Theorem for Unions states that, for all sufficiently large $k$, the sets $\mathscr U_k (H)$ are almost arithmeti…

2017-03-04math.ACmath.COmath.NT

Cohomological and Combinatorial Methods in the Study of Symbolic Powers and Equations defining Varieties

In this PhD thesis we will discuss some aspects in Commutative Algebra which have interactions with Algebraic Geometry, Representation Theory and Combinatorics. In particular, in the first chapter we will focus on understanding when certain cohomology modules vanish, a classical problem raised by Grothendieck. In the second chapter we will use local cohomology to study the connectedness behavior during a Groebner deformation and the arithmetical rank of certain varieties. In the third chapter, we will investigate the relations between the minors of a fixed size of a generic matrix by using too…

2011-05-27math.ACmath.AGmath.CO

Baxter Algebras and Differential Algebras

A Baxter algebra is a commutative algebra $A$ that carries a generalized integral operator. In the first part of this paper we review past work of Baxter, Miller, Rota and Cartier in this area and explain more recent work on explicit constructions of free Baxter algebras that extended the constructions of Rota and Cartier. In the second part of the paper we will use these explicit constructions to relate Baxter algebras to Hopf algebras and give applications of Baxter algebras to the umbral calculus in combinatorics.

2004-07-10math.ACmath.COmath.HO

Distributive Lattices, Bipartite Graphs and Alexander Duality

A certain squarefree monomial ideal $H_P$ arising from a finite partially ordered set $P$ will be studied from viewpoints of both commutative algebra and combinatorics. First, it is proved that the defining ideal of the Rees algebra of $H_P$ possesses a quadratic Gröbner basis. Thus in particular all powers of $H_P$ have linear resolutions. Second, the minimal free graded resolution of $H_P$ will be constructed explicitly and a combinatorial formula to compute the Betti numbers of $H_P$ will be presented. Third, by using the fact that the Alexander dual of the simplicial complex $Δ$ whose Stan…

2003-07-17math.ACmath.CO