arXiv subjects

Amir Mafi

Publications and source records attributed to Amir Mafi.

At least 19 recordsLinked to original sources

arXiv2026

On the Regularity of Dominant and Almost Complete Intersection Monomial Ideals

Let $R = k[x_1,\ldots,x_n]$ be a polynomial ring in $n$ variables over a field $k$, and let $I$ be a monomial ideal of $R$. If $I$ is an almost complete intersection, then we provide an explicit formula for the Castelnuovo-Mumford regularity of $I$ in terms of the powers of the dominant variables appearing in the regular sequence contained in $G(I)$ of length $|G(I)|-1$, where $G(I)$ is the set of minimal monomial generators of $I$. Furthermore, if $I$ is a dominant ideal or an almost complete intersection ideal, then we show that $\operatorname{reg}(\overline{I}) \leq \operatorname{reg}(I),$ where $\overline{I}$ denotes the integral closure of $I$. This provides a positive answer to the K\"uronya-Pintye conjecture for these two classes of monomial ideals. In addition, we give some examples to clarify these results.

math.AC↗

arXiv2025

Betti numbers and almost complete intersection monomial ideals

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and let $I$ be a monomial ideal of $R$. In this paper, we present an explicit formula for the Betti numbers of almost complete intersection monomial ideals, which enables a rapid construction of their minimal free resolutions. In addition, we characterize the Cohen-Macaulayness of these ideals and also we show the same result for dominant monomial ideals.

math.AC↗

arXiv2025

Sequentially Cohen-Macaulay and pretty clean monomial ideals

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be monomial ideal of $R$. In this paper, we show that if $I$ is a generic monomial ideal, then $R/I$ is pretty clean if and only if $R/I$ is sequentially Cohen-Macaulay. Furthermore, we prove that this equivalence remains unchanged for some special monomial ideals. Moreover, we provide an example that disproves the conjecture raised in \cite[p. 123]{S1} regarding generic monomial ideals.

math.AC↗

arXiv2025

Morse resolutions of monomial ideals and Betti splittings

We use discrete Morse theory to study free resolutions of monomial ideals in combination with splitting techniques. We establish the minimality of such pruned resolutions for several classes of ideals, including stable and linear quotient ideals. In particular, we unify classical constructions such as the Eliahou-Kervaire and Herzog-Takayama resolutions within the pruned resolution framework. Additionally, we introduce methods to reduce the minimality study of a pruned resolution for an ideal to that of a smaller subideal and present a variant of our pruned resolution for powers of monomial ideals.

math.AC↗

arXiv2024

Unmixed polymatroidal ideals

Let $R=K[x_1,\ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I$ be a polymatroidal ideal of $R$. In this paper, we provide a comprehensive classification of all unmixed polymatroidal ideals. This work addresses a question raised by Herzog and Hibi in [10]

math.AC↗

arXiv2023

An upper bound on stability of powers of matroidal ideals

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring in $n$ variables over a field $K$ and $I$ be a matroidal ideal of degree $d$. Let $\astab(I)$ and $\dstab(I)$ be the smallest integers $l$ and $k$, for which $\Ass(I^l)$ and $\depth(R/I^k)$ stabilize, respectively. In this paper, we show that $\astab(I),\dstab(I)\leq\min\{d,\ell(I)\}$, where $\ell(I)$ is the analytic spread of $I$. Furthermore, by a counterexample we give a negative answer to the conjecture of Herzog and Qureshi \cite{HQ} about stability of matroidal ideals.

math.AC↗

arXiv2023

A remark on sequentially Cohen-Macaulay monomial ideals

Let $R=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$. We show that if $G$ is a connected graph with a basic $5$-cycle $C$, then $G$ is a sequentially Cohen-Macaulay graph if and only if there exists a shedding vertex $x$ of $C$ such that $G\setminus x$ and $G\setminus N[x]$ are sequentially Cohen-Macaulay graphs. Furthermore, we study the sequentially Cohen-Macaulay and Castelnuovo-Mumford regularity of square-free monomial ideals in some special cases.

math.AC↗

arXiv2022

Polarizations of Artin monomial ideals

We show that any polarization of an Artin monomial ideal defines a triangulated ball. This proves a conjecture of A.Almousa, H.Lohne and the first author. Geometrically, polarizations of ideals containing $(x_1^{a_1}, \ldots, x_n^{a_n})$ define full-dimensional triangulated balls on the sphere which is the join of boundaries of simplices of dimensions $a_1-1, \cdots, a_n-1$. We prove that every full-dimensional Cohen-Macaulay sub-complex of this joined sphere is of this kind, and these balls are constructible. Such a triangulated ball has a dual cell complex which is a sub-complex of the product of simplices of dimensions $a_1-1, \cdots a_n-1$. We prove that this cell complex gives cellular minimal free resolution of this of the Alexander dual ideal of the triangulated ball. When the product of simplices is a hypercube, using these dual cell complexes we classify in a range examples all polarizations of the Artin monomial ideal. We also show that the squeezed balls of G.Kalai \cite{Ka} derive from polarizations of Artin monomial ideals.

math.AC↗

arXiv2022

Quasi-forest simplicial complexes and almost Cohen-Macaulay

In this paper we study the quasi-forest simplicial complexes and we define the concept of simplicial $k$-cycle (denoted by $\mathcal{S}_k$) and simplicial $k$-point (denoted by $\mathcal{P}_k$). We show that a simplicial complex $\Delta$ is quasi-forest if and only if it does not have any $\mathcal{P}_k$ and any $\mathcal{S}_k$ for $k\geq 3$. Furthermore we characterize almost Cohen-Macaulay quasi-forest simplicial complexes. In the end we show that the cycle graph $G=C_n$ is almost Cohen-Macaulay if and only if $n=3,4,5,6,7,8,9,11$.

math.AC↗

arXiv2022

Sequentially Cohen-Macaulay Co-Chordal Graphs: Structure and Projective Dimension

We introduce a class of chordal graphs called ($d_1$,$d_2$,$\dots$,$d_q$)-trees. A graph belongs to this class if and only if its clique complex is sequentially Cohen-Macaulay, providing a complete classification of all sequentially Cohen-Macaulay co-chordal graphs. This class also yields a classification of bi-sequentially Cohen-Macaulay graphs. We study the relationship between the projective dimension of a graph and its maximum vertex degree. We show that the projective dimension is always at least the maximum vertex degree, although this bound is not always tight, even for co-chordal graphs. However, equality holds when the graph is sequentially Cohen-Macaulay co-chordal or has a full vertex.

math.AC↗

arXiv2022

Strong persistence and associated prime of powers of monomial ideals

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal of degree $d\leq 2$. We show that $(I^{k+1}:I)=I^k$ for all $k\geq 1$ and we disprove a motivation question that was appeared in \cite[Question 2.51]{CHHV} by providing of a counterexample. Also, by this counterexample, we give a negative answer to the question that depth function of square-free monomial ideals are non-increasing.

math.AC↗

arXiv2022

On vertex decomposability and regularity of graphs

There are two motivation questions in \cite{MTS, MTS1} about Castelnuovo-Mumford regularity and vertex decomposable of simple graph $G$. In this paper, we disprove the questions by providing of two counterexamples.

math.CO↗

arXiv2022

Vertex decomposability and weakly polymatroidal ideals

Let $K$ be a field and $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$. Let $\Delta$ be a simplicial complex on $n$ vertices and $I=I_{\Delta}$ be its Stanley-Reisner ideal. In this paper, we show that if $I$ is a matroidal ideal then the following conditions are equivalent: $(i)$ $\Delta$ is sequentially Cohen-Macaulay; $(ii)$ $\Delta$ is shellable; $(iii)$ $\Delta$ is vertex decomposable. Also, if $I$ is a minimally generated by $u_1,\ldots,u_s$ such that $s\leq 3$ or ${\rm supp}(u_i)\cup {\rm supp}(u_j)=\{x_1,\ldots,x_n\}$ for all $i\neq j$, then $\Delta$ is vertex decomposable. Furthermore, we prove that if $I$ is a monomial ideal of degree $2$ then $I$ is weakly polymatroidal if and only if $I$ has linear quotients if and only if $I$ is vertex splittable.

math.AC↗

arXiv2021

Almost Cohen-Macaulay bipartite graphs and connected in codimension two

In this paper we study almost Cohen-Macaulay bipartite graphs. Furthermore, we prove that if $G$ is almost Cohen-Macaulay bipartite graph with at least one vertex of positive degree, then there is a vertex of $\deg(v) \leq 2$. In particular, if $G$ is an almost Cohen-Macaulay bipartite graph and $u$ is a vertex of degree one of $G$ and $v$ its adjacent vertex, then $G\setminus\{v\}$ is almost Cohen-Macaulay. Also, we show that an unmixed Ferrers graph is almost Cohen-Macaulay if and only if it is connected in codimension two. Moreover, we give some examples.

math.AC↗

arXiv2021

A note on stability properties of powers of polymatroidal ideals

Let $I$ be a matroidal ideal of degrre $d$ of a polynomial ring $R=K[x_1,...,x_n]$, where $K$ is a field. Let astab$(I)$ and dstab$(I)$ be the smallest integer $n$ for which Ass$(I^n)$ and depth$(I^n)$ stabilize, respectively. In this paper, we show that astab$(I)=1$ if and only if dstab$(I)=1$. Moreover, we prove that if $d=3$, then ${\rm astab}(I)={\rm dstab}(I)$. Furthermore, we show that if $I$ is an almost square-free Veronese type ideal of degree $d$, then ${\rm astab}(I)={\rm dstab}(I)=\lceil\frac{n-1}{n-d}\rceil$.

math.AC↗

arXiv2021

Integral closure and Hilbert series of a special monomial ideal

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and let $M_{n,t}=(x^{e_1},\ldots, x^{e_n})$ be a monomial ideal of $R$, where $x^{e_i}=x_1^t\ldots x_{i-1}^tx_{i+1}^t\ldots x_n^t$. We study the unmixedness of its integral closure. Furthermore, we compute the Hilbert series of this ideal and we show that this ideal is Freiman.

math.AC↗

arXiv2021

A note on almost Cohen-Macaulay monomial ideals

Let $R = k[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $k$ and let $I$ be a monomial ideal of $R$. In this paper, we study almost Cohen-Macaulay simplicial complex. Moreover, we characterize the almost Cohen-Macaulay polymatroidal Veronese type and transversal polymatroidal ideals and furthermore we give some examples.

math.AC↗

arXiv2019

On the Hilbert coefficients, depth of associated graded rings and reduction numbers

Let $(R,\mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J=(x_1,...,x_d)$ a minimal reduction of $I$. We show that if $J_{d-1}=(x_1,...,x_{d-1})$ and $\sum\limits_{n=1}^\inftyλ{({I^{n+1}\cap J_{d-1}})/({J{I^n} \cap J_{d-1}})=i}$ where i=0,1, then depth $G(I)\geq{d-i-1}$. Moreover, we prove that if $e_2(I) = \sum_{n=2}^\infty (n-1) λ(I^n/JI^{n-1})-2;$ or if $I$ is integrally closed and $e_2(I) = \sum_{n=2}^\infty (n-1)λ({I^{n}}/JI^{n-1})-i$ where $i=3,4$, then $e_1(I) = \sum_{n=1}^\infty λ(I^n / JI^{n-1})-1.$ In addition, we show that $r(I)$ is independent. Furthermore, we study the independence of $r(I)$ with some other conditions.

math.AC↗