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Azhar Farooq

Publications and source records attributed to Azhar Farooq.

6 recordsLinked to original sources
arXiv2026

Some Properties of Path and Closed Neighborhood Ideals

In this paper, we investigate algebraic properties of two classes of monomial ideals associated with graphs, namely path ideals and closed neighborhood ideals. Our primary focus is on the strong persistence property and several homological invariants, including depth and the $v$-number. We first prove that the path ideal of length $n-1$ associated with a path graph on $n\geq4$ vertices attached to an arbitrary graph satisfies the strong persistence property. We then study closed neighborhood ideals and establish the strong persistence property for graphs containing a leaf vertex. Furthermore, we identify a broader class of graphs satisfying this property, namely those whose vertex set admits a decomposition $V(G)=C\sqcup\{v\}\sqcup X,$ where $C$ is a nonempty clique, every vertex of $C$ is adjacent to each vertex of $X\cup\{v\}$, and the vertex $v$ has no neighbors in $X$. In addition, we investigate algebraic invariants of ideals associated with Helm graphs. Specifically, we compute the depth of the edge ideal and determine the Krull dimension, the depth, and the $v$-number of the closed neighborhood ideal of the Helm graph. These results contribute to a deeper understanding of the interplay between graph-theoretic structures and the algebraic properties of their associated monomial ideals.

math.AC↗
arXiv2026

Rota Baxter Operators on Truncated Polynomial Algebras

Let K be a field of characteristic zero, and let m=(x_1,...,x_n)) be a maximal ideal of the polynomial ring K[x_1,...,x_n]. We classify all Rota--Baxter operators of weights zero and one on the truncated polynomial algebra R=K[x_1,\dots,x_n]/m^2. For weight zero, we prove that the Rota--Baxter operators are precisely the linear maps P satisfying P^2=0 and Image(P) \subset m/m^2. For nonzero weight, a standard rescaling reduces the classification to weight one. In this case, the operators split into two disjoint families according to the value of P(1)\in{0,-1}. On the maximal ideal m/m^2, such operators induce an endomorphism L satisfying L^2 + L = 0), equivalently, -L is idempotent. We further show that each family is isomorphic to the variety of idempotent matrices.

math.AC↗
arXiv2022

Image closure of symmetric wide-matrix varieties

Let $X$ be an affine scheme of $k \times \mathbb{N}$-matrices and $Y$ be an affine scheme of $\mathbb{N} \times \cdots \times \mathbb{N}$-dimensional tensors. The group Sym$(\mathbb{N})$ acts naturally on both $X$ and $Y$ and on their coordinate rings. We show that the Zariski closure of the image of a Sym$(\mathbb{N})$-equivariant morphism of schemes from $X$ to $Y$ is defined by finitely many Sym$(\mathbb{N})$-orbits in the coordinate ring of $Y$. Moreover, we prove that the closure of the image of this map is Sym$(\mathbb{N})$-Noetherian, that is, every descending chain of Sym$(\mathbb{N})$-stable closed subsets stabilizes.

math.AG↗
arXiv2022

Sym-Noetherianity for powers of GL-varieties

Much recent literature concerns finiteness properties of infinite-dimensional algebraic varieties equipped with an action of the infinite symmetric group, or of the infinite general linear group. In this paper, we study a common generalisation in which the product of both groups acts on infinite-dimensional spaces, and we show that these spaces are topologically Noetherian with respect to this action.

math.AG↗
arXiv2022

Components of symmetric wide-matrix varieties

We show that if X_n is a variety of cxn-matrices that is stable under the group Sym([n]) of column permutations and if forgetting the last column maps X_n into X_{n-1}, then the number of Sym([n])-orbits on irreducible components of X_n is a quasipolynomial in n for all sufficiently large n. To this end, we introduce the category of affine FI^op-schemes of width one, review existing literature on such schemes, and establish several new structural results about them. In particular, we show that under a shift and a localisation, any width-one FI^op-scheme becomes of product form, where X_n=Y^n for some scheme Y in affine c-space. Furthermore, to any FI^op-scheme of width one we associate a component functor from the category FI of finite sets with injections to the category PF of finite sets with partially defined maps. We present a combinatorial model for these functors and use this model to prove that Sym([n])-orbits of components of X_n, for all n, correspond bijectively to orbits of a groupoid acting on the integral points in certain rational polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on quasipolynomiality of lattice point counts, this yields our Main Theorem.

math.AC↗