The Stanley regularity of complete intersections and ideals of mixed products

2016 ◽  
Vol 16 (07) ◽  
pp. 1750122
Author(s):  
Lizhong Chu ◽  
V. H. Jorge Pérez
Keyword(s):  
Complete Intersection ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Complete Intersections

Let [Formula: see text] be a polynomial ring over a field [Formula: see text] and [Formula: see text] a monomial ideal. We give some inequalities on Stanley regularity of monomial ideals. As consequences, we prove that [Formula: see text] and [Formula: see text] hold in the following cases: (1) [Formula: see text] is a complete intersection; (2) [Formula: see text] is an ideal of mixed products.

scholarly journals Monomial multiplicities in explicit form

2019 ◽  
Vol 19 (09) ◽  
pp. 2050181
Author(s):  
Guillermo Alesandroni
Keyword(s):  
Explicit Formula ◽  
Explicit Form ◽  
Complete Intersection ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
The Other ◽  
Complete Intersections ◽  
New Class ◽  
Other Hand ◽  
The Family

Denote by [Formula: see text] a polynomial ring over a field, and let [Formula: see text] be a monomial ideal of [Formula: see text]. If [Formula: see text], we prove that the multiplicity of [Formula: see text] is given by [Formula: see text] On the other hand, if [Formula: see text] is a complete intersection, and [Formula: see text] is an almost complete intersection, we show that [Formula: see text] We also introduce a new class of ideals that extends the family of monomial complete intersections and that of codimension 1 ideals, and give an explicit formula for their multiplicity.

scholarly journals Stanley’s conjecture for critical ideals

2011 ◽  
Vol 48 (2) ◽  
pp. 220-226
Author(s):  
Azeem Haider ◽  
Sardar Khan
Keyword(s):  
Polynomial Ring ◽  
Hilbert Function ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Stanley Depth

Let S = K[x1,…,xn] be a polynomial ring in n variables over a field K. Stanley’s conjecture holds for the modules I and S/I, when I ⊂ S is a critical monomial ideal. We calculate the Stanley depth of S/I when I is a canonical critical monomial ideal. For non-critical monomial ideals we show the existence of a Stanley ideal with the same depth and Hilbert function.

scholarly journals On Monomial Golod Ideals

2020 ◽  
Author(s):  
Hailong Dao ◽  
Alessandro De Stefani
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Complete Characterization ◽  
Monomial Ideals

Abstract We study ideal-theoretic conditions for a monomial ideal to be Golod. For ideals in a polynomial ring in three variables, our criteria give a complete characterization. Over such rings, we show that the product of two monomial ideals is Golod.

scholarly journals ON THE CONJECTURE OF VASCONCELOS FOR ARTINIAN ALMOST COMPLETE INTERSECTION MONOMIAL IDEALS

2019 ◽  
pp. 1-15
Author(s):  
KUEI-NUAN LIN ◽  
YI-HUANG SHEN
Keyword(s):  
Complete Intersection ◽  
Monomial Ideal ◽  
Short Note ◽  
Monomial Ideals ◽  
Rees Algebra

In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen–Macaulay.

ON THE STANLEY DEPTH AND SIZE OF MONOMIAL IDEALS

2017 ◽  
Vol 59 (3) ◽  
pp. 705-715
Author(s):  
S. A. SEYED FAKHARI
Keyword(s):  
Lower Bound ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
Recursive Formula ◽  
Monomial Ideals ◽  
Stanley Depth

AbstractLet $\mathbb{K}$ be a field and S = ${\mathbb{K}}$[x1, . . ., xn] be the polynomial ring in n variables over the field $\mathbb{K}$. For every monomial ideal I ⊂ S, we provide a recursive formula to determine a lower bound for the Stanley depth of S/I. We use this formula to prove the inequality sdepth(S/I) ≥ size(I) for a particular class of monomial ideals.

On the reduction numbers of monomial ideals

2019 ◽  
Vol 19 (10) ◽  
pp. 2050201
Author(s):  
Ibrahim Al-Ayyoub
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Upper Bounds ◽  
Integral Closure ◽  
Monomial Ideals ◽  
Explicit Method ◽  
Generating Set ◽  
Ideal Formula ◽  
Reduction Number ◽  
The Ideal

Let [Formula: see text] be a monomial ideal in a polynomial ring with two indeterminates over a field. Assume [Formula: see text] is contained in the integral closure of some ideal that is generated by two elements from the generating set of [Formula: see text]. We produce sharp upper bounds for each of the reduction number and the Ratliff–Rush reduction number of the ideal [Formula: see text]. Under certain hypotheses, we give the exact values of these reduction numbers, and we provide an explicit method for obtaining these sharp upper bounds.

Partial Castelnuovo–Mumford regularities of sums and intersections of powers of monomial ideals

2010 ◽  
Vol 149 (2) ◽  
pp. 229-246 ◽  
Author(s):  
LÊ TUÂN HOA ◽  
TRÂN NAM TRUNG
Keyword(s):  
Asymptotic Behavior ◽  
Linear Function ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
Monomial Ideals ◽  
Symbolic Powers ◽  
Integral Closures

AbstractLet I, I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp be monomial ideals of a polynomial ring R = K[X1,. . ., Xr] and Ln = I+∩jIn1j + ⋅ ⋅ ⋅ + ∩jIpjn. It is shown that the ai-invariant ai(R/Ln) is asymptotically a quasi-linear function of n for all n ≫ 0, and the limit limn→∞ad(R/Ln)/n exists, where d = dim(R/L1). A similar result holds if I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp are replaced by their integral closures. Moreover all limits $\lim_{n\to\infty} a_i(R/(\cap_j \overline{I_{1j}^n} + \cdots + \cap_j \overline{I_{pj}^n}))/n $ also exist.As a consequence, it is shown that there are integers p > 0 and 0 ≤ e ≤ d = dim R/I such that reg(In) = pn + e for all n ≫ 0 and pn ≤ reg(In) ≤ pn + d for all n > 0 and that the asymptotic behavior of the Castelnuovo–Mumford regularity of ordinary symbolic powers of a square-free monomial ideal is very close to a linear function.

scholarly journals Free resolution of powers of monomial ideals and Golod rings

2017 ◽  
Vol 120 (1) ◽  
pp. 59 ◽  
Author(s):  
N. Altafi ◽  
N. Nemati ◽  
S. A. Seyed Fakhari ◽  
S. Yassemi
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Monomial Order ◽  
Golod Ring ◽  
Linear Quotients ◽  
The Ideal

Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this paper we present a criterion for componentwise linearity of powers of monomial ideals. In particular, we prove that if a square-free monomial ideal $I$ contains no variable and some power of $I$ is componentwise linear, then $I$ satisfies the gcd condition. For a square-free monomial ideal $I$ which contains no variable, we show that $S/I$ is a Golod ring provided that for some integer $s\geq 1$, the ideal $I^s$ has linear quotients with respect to a monomial order.

scholarly journals Invariant subrings which are complete intersections, I: (Invariant subrings of finite Abelian groups)

1980 ◽  
Vol 77 ◽  
pp. 89-98 ◽  
Author(s):  
Keiichi Watanabe
Keyword(s):  
Abelian Group ◽  
Complete Intersection ◽  
Polynomial Ring ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Subgroup ◽  
Complete Intersections ◽  
Complex Numbers ◽  
Finite Abelian Groups

Let G be a finite subgroup of GL(n, C) (C is the field of complex numbers). Then G acts naturally on the polynomial ring S = C[X1, …, Xn]. We consider the followingProblem. When is the invariant subring SG a complete intersection?In this paper, we treat the case where G is a finite Abelian group. We can solve the problem completely. The result is stated in Theorem 2.1.

scholarly journals Minimum distance functions of complete intersections

2018 ◽  
Vol 17 (11) ◽  
pp. 1850204 ◽  
Author(s):  
José Martínez-Bernal ◽  
Yuriko Pitones ◽  
Rafael H. Villarreal
Keyword(s):  
Lower Bound ◽  
Complete Intersection ◽  
Polynomial Ring ◽  
Distance Function ◽  
Coding Theory ◽  
Minimum Distance ◽  
Distance Functions ◽  
Complete Intersections ◽  
Dimension One ◽  
Sharp Lower Bound

We study the minimum distance function of a complete intersection graded ideal in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we use the footprint function to give a sharp lower bound for the minimum distance function. Then we show some applications to coding theory.

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