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Abed Abedelfatah

Publications and source records attributed to Abed Abedelfatah.

8 recordsLinked to original sources
arXiv2026

Two Infinite Families of Regular Sequences of Power Sums in Three Variables

Let $$ S=\mathbb{C}[x,y,z],\qquad p_m=x^m+y^m+z^m. $$ For every $r\geq1$, we prove that $$ p_r,\ p_{r+1},\ p_{3r+1} \qquad\text{and}\qquad p_r,\ p_{r+1},\ p_{3r+2} $$ are regular sequences exactly when $r\not\equiv1\pmod3$. This proves the Conca--Krattenthaler--Watanabe prediction for two infinite families. The proof is elementary. Four simple reductions modulo $(p_r,p_{r+1})$ reduce the problem to the two equations $x+y+z=0$ and $xy+xz+yz=0$.

math.AC↗
arXiv2026

Quadratic Ideals in Six Variables and the Eisenbud--Green--Harris Conjecture

In this paper, we study the Eisenbud--Green--Harris (EGH) conjecture for ideals generated by quadrics. We establish a sharp lower bound for the dimension of the cubic component of an ideal generated by a regular sequence of six quadrics and two additional quadrics in six variables. Furthermore, we prove the Eisenbud--Green--Harris conjecture for almost complete intersections of quadrics in six variables.

math.AC↗
arXiv2023

On the subadditivity condition of edge ideal

Let $S=K[x_1,\ldots,x_n]$, where $K$ is a field, and $t_i(S/I)$ denotes the maximal shift in the minimal graded free $S$-resolution of the graded algebra $S/I$ at degree $i$, where $I$ is an edge ideal. In this paper, we prove that if $t_b(S/I)\geq \lceil \frac{3b}{2} \rceil$ for some $b\geq 0$, then the subadditivity condition $t_{a+b}(S/I)\leq t_a(S/I)+t_b(S/I)$ holds for all $a\geq 0$. In addition, we prove that $t_{a+4}(S/I)\leq t_a(S/I)+t_4(S/I)$ for all $a\geq 0$ (the case $b=0,1,2,3$ is known). We conclude that if the projective dimension of $S/I$ is at most $9$, then $I$ satisfies the subadditivity condition.

math.AC↗
arXiv2020

Some results on the subadditivity condition of syzygies

Among other results, we prove that if $I$ is a monomial ideal of $S=K[x_1,\ldots,x_n]$, where $K$ is a field, and $a\geq b-1\geq0$ are integers such that $a+b\leq\mathrm{proj~dim}(S/I)$, then $$t_{a+b}\leq t_a+t_1+t_2+\cdots+t_b-\frac{b(b-1)}{2},$$ where $t_1,t_2,\dots$ are the maximal shifts in the minimal graded free $S$-resolution of $S/I$.

math.AC↗
arXiv2016

On vanishing patterns in $j$-strands of edge ideals

We consider two problems regarding vanishing patterns in the Betti table of edge ideals $I$ in polynomial algebra $S$. First, we show that the $j$-strand is connected if $j=3$ (for $j=2$ this is easy and known), and give examples where the $j$-strand is not connected for any $j>3$. Next, we apply our result on strand connectivity to establish the subadditivity conjecture for edge ideals, $t_{a+b}\leq t_a+t_b$, in case $b=2,3$ (the case $b=1$ is known). Here $t_i$ stands for the maximal shifts in the minimal free $S$-resolution of $S/I$

math.AC↗
arXiv2015

Hilbert functions of monomial ideals containing a regular sequence

Let $M$ be an ideal in $K[x_1,...,x_n]$ ($K$ is a field) generated by products of linear forms and containing a homogeneous regular sequence of some length. We prove that ideals containing $M$ satisfy the Eisenbud-Green-Harris conjecture and moreover prove that the Cohen-Macaulay property is preserved. We conclude that monomial ideals satisfy this conjecture. We obtain that $h$-vector of Cohen-Macaulay simplicial complex $Δ$ is the $h$-vector of Cohen-Macaulay $(a_1-1,...,a_t-1)$-balanced simplicial complex where $t$ is the height of the Stanley-Reisner ideal of $Δ$ and $(a_1,...,a_t)$ is the type of some regular sequence contained in this ideal.

math.AC↗
arXiv2012

On the Eisenbud-Green-Harris Conjecture

It has been conjectured by Eisenbud, Green and Harris that if $I$ is a homogeneous ideal in $k[x_1,...,x_n]$ containing a regular sequence $f_1,...,f_n$ of degrees $°(f_i)=a_i$, where $2\leq a_1\leq ... \leq a_n$, then there is a homogeneous ideal $J$ containing $x_1^{a_1},...,x_n^{a_n}$ with the same Hilbert function. In this paper we prove the Eisenbud-Green-Harris conjecture when $f_i$ splits into linear factors for all $i$.

math.AC↗