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Stanley depth of powers of the edge ideal of a forest
, Article Proceedings of the American Mathematical Society ; Volume 141, Issue 10 , 2013 , Pages 3327-3336 ; 00029939 (ISSN) ; Seyed Fakhari, S. A ; Yassemi, S ; Sharif University of Technology
2013
Abstract
Let K be a field and S = K[x1,...,xn] be the polynomial ring in n variables over the field K. Let G be a forest with p connected components G1,...,Gp and let I = I(G) be its edge ideal in S. Suppose that di is the diameter of Gi, 1 ≤ i ≤ p, and consider d = max{di I 1 ≤ i ≤ p}. Morey has shown that for every t ≥ 1, the quantity max is a lower bound for depth(S/It). In this paper, we show that for every t ≥ 1, the mentioned quantity is also a lower bound for sdepth(S/It). By combining this inequality with Burch's inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem
On the Stanley depth of weakly polymatroidal ideals
, Article Archiv der Mathematik ; Volume 100, Issue 2 , 2013 , Pages 115-121 ; 0003889X (ISSN) ; Seyed Fakhari, S. A ; Yassemi, S ; Sharif University of Technology
2013
Abstract
Let K be a field and S = K[x1,...,xn] be the polynomial ring in n variables over the field K. In this paper, it is shown that Stanley's conjecture holds for I and S/I if I is a product of monomial prime ideals or I is a high enough power of a polymatroidal or a stable ideal generated in a single degree
On the h-triangles of sequentially (S r) simplicial complexes via algebraic shifting
, Article Arkiv for Matematik ; Volume 51, Issue 1 , 2013 , Pages 185-196 ; 00042080 (ISSN) ; Seyed Fakhari, S. A ; Yassemi, S ; Sharif University of Technology
2013
Abstract
Recently, Haghighi, Terai, Yassemi, and Zaare-Nahandi introduced the notion of a sequentially (Sr) simplicial complex. This notion gives a generalization of two properties for simplicial complexes: being sequentially Cohen-Macaulay and satisfying Serre's condition (Sr). Let Δ be a (d-1)-dimensional simplicial complex with Γ(Δ) as its algebraic shifting. Also let (hi,j(Δ))0≤j≤i≤d be the h-triangle of Δ and (hi,j(Γ(Δ)))0≤j≤i≤d be the h-triangle of Γ(Δ). In this paper, it is shown that for a Δ being sequentially (Sr) and for every i and j with 0≤j≤i≤r-1, the equality hi,j(Δ)=hi,j(Γ(Δ)) holds true
On the h-vector of a simplicial complex with Serre's condition
, Article Journal of Pure and Applied Algebra ; Volume 216, Issue 1 , January , 2012 , Pages 91-94 ; 00224049 (ISSN) ; Pournaki, M. R ; Seyed Fakhari, S. A ; Yassemi, S
2012
Abstract
Let δ be a (d-1)-dimensional simplicial complex and let h(δ)=(h0,h1,...,hd) be its h-vector. A recent result of Murai and Terai guarantees that if δ satisfies Serre's condition (Sr), then (h0,h1,...,hr) is an M-vector and hr+hr+1+...+hd is nonnegative. In this article, we extend the result of Murai and Terai by giving r extra necessary conditions. More precisely, we prove that if δ satisfies Serre's condition (Sr), then iihr+i+1ihr+1+...+i+d-rihd, 0≤i≤r≤d, are all nonnegative
New Classes of Set-theoretic Complete Intersection Monomial Ideals
, Article Communications in Algebra ; Volume 43, Issue 9 , Jun , 2015 , Pages 3920-3924 ; 00927872 (ISSN) ; Seyed Fakhari, S. A ; Yassemi, S ; Sharif University of Technology
Taylor and Francis Inc
2015
Abstract
Let Δ be a simplicial complex and χ be an s-coloring of Δ. Biermann and Van Tuyl have introduced the simplicial complex Δχ . As a corollary of Theorems 5 and 7 in their 2013 article, we obtain that the Stanley–Reisner ring of Δχ over a field is Cohen–Macaulay. In this note, we generalize this corollary by proving that the Stanley–Reisner ideal of Δχ over a field is set-theoretic complete intersection. This also generalizes a result of Macchia
The clique numbers of regular graphs of matrix algebras are finite
, Article Linear Algebra and Its Applications ; Volume 431, Issue 10 , 2009 , Pages 1715-1718 ; 00243795 (ISSN) ; Jamaali, M ; Seyed Fakhari, S. A ; Sharif University of Technology
2009
Abstract
Let F be a field, char (F) ≠ 2, and S ⊆ GLn (F), where n is a positive integer. In this paper we show that if for every distinct elements x, y ∈ S, x + y is singular, then S is finite. We conjecture that this result is true if one replaces field with a division ring. © 2009 Elsevier Inc. All rights reserved
Survey article: Simplicial complexes satisfying Serre's condition: A survey with some new results
, Article Journal of Commutative Algebra ; Vol. 6, issue. 4 , 2014 , p. 455-483 ; Seyed Fakhari, S. A ; Terai, N ; Yassemi, S ; Sharif University of Technology
Abstract
The problem of finding a characterization of Cohen-Macaulay simplicial complexes has been studied intensively by many authors. There are several attempts at this problem available for some special classes of simplicial complexes satisfying some technical conditions. This paper is a survey, with some new results, of some of these developments. The new results about simplicial complexes with Serre's condition are an analogue of the known results for Cohen-Macaulay simplicial complexes
Combinatorics comes to the rescue: h-vectors in commutative algebra
, Article Mathematical Intelligencer ; 2018 ; 03436993 (ISSN) ; Pournaki, M.R ; Seyed Fakhari, S.A ; Yassemi, S ; Sharif University of Technology
Springer New York LLC
2018
Combinatorics comes to the rescue: H-vectors in commutative algebra
, Article Mathematical Intelligencer ; Volume 41, Issue 1 , 2019 , Pages 16-21 ; 03436993 (ISSN) ; Pournaki, M. R ; Seyed Fakhari, S. A ; Yassemi, S ; Sharif University of Technology
Springer New York LLC
2019
Cohen-Macaulayness and Limit Behavior of Depth for Powers of Cover Ideals
, Article Communications in Algebra ; Volume 43, Issue 1 , Aug , 2015 , Pages 143-157 ; 00927872 (ISSN) ; Pournaki, M. R ; Seyed Fakhari, S. A ; Terai, N ; Yassemi, S ; Sharif University of Technology
Taylor and Francis Inc
2015
Abstract
Let K{double-struck} be a field, and let R = K{double-struck}[x1,.., xn] be the polynomial ring over K{double-struck} in n indeterminates x1,.., xn. Let G be a graph with vertex-set {x1,.., xn}, and let J be the cover ideal of G in R. For a given positive integer k, we denote the kth symbolic power and the kth bracket power of J by J(k) and J[k], respectively. In this paper, we give necessary and sufficient conditions for R/Jk, R/J (k), and R/J [k] to be Cohen-Macaulay. We also study the limit behavior of the depths of these rings
A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra
, Article Research in Mathematical Sciences ; Volume 9, Issue 2 , 2022 ; 25220144 (ISSN) ; Pournaki, M. R ; Seyed Fakhari, S. A ; Terai, N ; Yassemi, S ; Sharif University of Technology
Springer Science and Business Media Deutschland GmbH
2022
Abstract
A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG