arXiv subjects

Louiza Fouli

Publications and source records attributed to Louiza Fouli.

At least 19 recordsLinked to original sources

arXiv2026

Algebraic invariants of special fiber rings of ladder determinantal modules

We provide explicit formulas for key invariants of special fiber rings of ladder determinantal modules, that is, modules that are direct sums of ideals of maximal minors of a ladder matrix. Our results are given in terms of the combinatorial data of the associated ladder matrix. In particular, we compute its dimension, regularity, $a$-invariant, and multiplicity, which via \textsc{Sagbi} degeneration coincide with those of Hibi rings associated to a distributive lattice. Then, via Gröbner degeneration these calculations are reduced to those of quotients of polynomial rings by monomial ideals. Our formula for the multiplicity of the special fiber ring of these ladder determinantal modules is obtained by counting the number of standard skew Young tableaux associated to a certain skew partition, and so provides a natural generalization of the classical formula for the degree of the Grassmannian.

math.AC↗

arXiv2026

Graded Betti numbers of graded Möbius algebras of uniform matroids

Graded Möbius algebras were a key tool in the proof of the Dowling-Wilson Top Heavy Conjecture. They are commutative algebras whose Hilbert functions recover the Whitney numbers of the second kind, i.e. the number of flats of a given rank. The graded Betti numbers of the defining ideal of a graded Möbius algebra refine the Hilbert function and describe its minimal free resolution. In this paper we derive precise formulas for the graded Betti numbers of the defining ideals of graded Möbius algebra for any uniform matroid. We also study when the graded Möbius algebra of an arbitrary matroid is linearly presented.

math.AC↗

arXiv2026

Asymptotic Resurgence of Facet ideals of Graphic Matroids

Our main results are an upper bound on the asymptotic resurgence of the facet ideal of a graphic matroid in terms of the number of vertices and a lower bound in terms of the circumference. These bounds coincide for Hamiltonian graphs, which form the majority of graphs on $n$ vertices as $n$ tends to infinity. For simple $2$-connected graphs on up to nine vertices, we compute in Sage that the asymptotic resurgence of the facet ideal of a non-Hamiltonian graphic matroid is given either by our upper or lower bound.

math.AC↗

arXiv2026

Asymptotic Resurgence of Facet and Stanley-Reisner ideals of Matroids

Matroid configurations -- introduced by Geramita, Harbourne, Migliore, and Nagel -- are projective varieties which generalize so-called \textit{star configurations} and whose defining ideals are obtained by appropriately specializing the Stanley-Reisner ideal of a matroid. Motivated by this connection, we study the asymptotic resurgence of the Stanley-Reisner ideals of matroids. A result of Villareal shows that it is equivalent to study the asymptotic resurgence of facet ideals. We prove a formula for the asymptotic resurgence of the facet ideal of a matroid in terms of the Waldschmidt constant of facet ideals of the contractions of the matroid. As a consequence, we show that asymptotic resurgence respects the weak order on matroids of the same rank. Therefore, the asymptotic resurgence of the facet ideal of a given matroid is bounded above by the asymptotic resurgence of the facet ideal of a so-called \textit{almost-uniform} matroid of the same rank, which we compute explicitly. Guardo, Harbourne, and Van Tuyl showed that the asymptotic resurgence of an ideal is bounded below by the ratio of the initial degree of the ideal by its Waldschmidt constant. We prove that this lower bound is an equality for facet ideals of many classes of matroids, including matroids of rank $k$ on a ground set of size $n\ge 2k$ whose dual is paving, perfect matroid designs, and sparse paving matroids arising from Steiner systems. For the latter two classes, we explicitly compute the asymptotic resurgence.

math.AC↗

arXiv2026

Regular sequences of linear forms on monomial ideals

In this paper we establish a means of using the combinatorics associated to a general monomial ideal $I$ in a polynomial ring $R$ to find a regular sequence of linear forms on $R/I$. The sequence of linear forms provides an effective lower bound on ${\rm{depth}}(R/I)$. When $I$ is the edge ideal of a graph, we provide conditions under which this bound is an equality, allowing the realization of the depth via a regular sequence of linear polynomials. In addition, we explicitly describe the minimal primes of $(I, f_1, \ldots, f_q)$, when $f_1, \ldots, f_q$ are homogeneous polynomials of degree one with pairwise disjoint support and $I$ is any monomial ideal. Finally, we propose a conjecture on the form of all associated primes of the ideal $(I, f_1, \ldots, f_q)$, when $I$ is the edge ideal of a graph and $f_i$ are disjoint stars on $I$.

math.AC↗

arXiv2026

Generalized Hamming weights and symbolic powers of Stanley-Reisner ideals of matroids

It is well-known that the first generalized Hamming weight of a linear code, more commonly called \textit{the minimum distance} of the linear code, corresponds to the initial degree of the Stanley-Reisner ideal of the matroid of the dual code. Our starting point in this paper is a generalization of this fact -- namely, the $r$-th generalized Hamming weight of a matroid is the smallest degree of a squarefree monomial in the $r$-th symbolic power of the Stanley-Reisner ideal of the matroid (in the appropriate range for $r$). We show that the squarefree monomials in successive symbolic powers of the Stanley-Reisner ideal of a matroid suffice to describe all symbolic powers of the Stanley-Reisner ideal. Hence, we provide explicit expressions for initial degree statistics of symbolic powers of the Stanley-Reisner ideal of a matroid in terms of its generalized Hamming weights. A key aspect of our approach is a careful study of duality. If the generalized Hamming weights of a matroid and its dual are both subadditive, we prove a simple expression for the initial degree of every symbolic power of the Stanley-Reisner ideal of the matroid, which closely mirrors that of a uniform matroid. This has unexpectedly far-reaching consequences - we prove the generalized Hamming weights of a matroid and its dual are both subadditive for many interesting classes of matroids and codes, including sparse paving matroids, perfect matroid designs, matroids arising from Steiner systems, first-order affine and projective Reed-Muller codes, constant weight codes, Griesmer codes, and perfect codes. As an application, we study the resurgence and asymptotic resurgence of the matroid configurations introduced by Geramita-Harbourne-Migliore-Nagel. In particular, we explicitly compute the asymptotic resurgence of a matroid configuration of points arising from a perfect matroid design.

math.AC↗

arXiv2026

Gorenstein Special Fiber Rings of Ladder Determinantal Modules

A ladder determinantal module is an arbitrary direct sum of ideals of maximal minors of a generic ladder matrix. In this article, we give necessary and sufficient conditions for the special fiber ring of such modules to be Gorenstein. These conditions are expressed in terms of data obtained from the underlying matrix.

math.AC↗

arXiv2023

The core of monomial ideals

The core of an ideal is defined as the intersection of all of its reductions. In this paper we provide an explicit description for the core of a monomial ideal $I$ satisfying certain residual conditions, showing that ${\rm core}(I)$ coincides with the largest monomial ideal contained in a general reduction of $I$. We prove that the class of lex-segment ideals satisfies these residual conditions and study the core of lex-segment ideals generated in one degree. For monomial ideals that do not necessarily satisfy the residual conditions and that are generated in one degree, we conjecture an explicit formula for the core, and make progress towards this conjecture.

math.AC↗

arXiv2022

Regular Sequences On Squares Of Monomial Ideals

We use initially regular sequences that consist of linear sums to explore the depth of $R/I^2$, when $I$ is a monomial ideal in a polynomial ring $R$. We give conditions under which these linear sums form regular or initially regular sequences on $R/I^2$. We then obtain a criterion for when $\depth R/I^2>1$ and a lower bound on $\depth R/I^2$.

math.AC↗

arXiv2022

Residual Intersections and Core of Modules

We introduce the notion of residual intersections of modules and prove their existence. We show that projective dimension one modules have Cohen-Macaulay residual intersections, namely they satisfy the relevant Artin-Nagata property. We then establish a formula for the core of orientable modules satisfying certain homological conditions, extending previous results of Corso, Polini, and Ulrich on the core of projective one modules. Finally, we provide examples of classes of modules that satisfy our assumptions.

math.AC↗

arXiv2021

Rees algebras of sparse determinantal ideals

We determine the defining equations of the Rees algebra and of the special fiber ring of the ideal of maximal minors of a $2\times n$ sparse matrix. We prove that their initial algebras are ladder determinantal rings. This allows us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rational singularities in characteristic zero and are F-rational in positive characteristic.

math.AC↗

arXiv2019

Depth of powers of squarefree monomial ideals

We derive two general bounds for the depths of powers of squarefree monomial ideals corresponding to hyperforests. These bounds generalize known bounds for the depths of squarefree monomial ideals, which were given in terms of the edgewise domination number of the corresponding hypergraphs and the lengths of initially regular sequences with respect to the ideals.

math.AC↗

arXiv2019

Initially regular sequences and depths of ideals

For an arbitrary ideal $I$ in a polynomial ring $R$ we define the notion of initially regular sequences on $R/I$. These sequences share properties with regular sequences. In particular, the length of an initially regular sequence provides a lower bound for the depth of $R/I$. Using combinatorial information from the initial ideal of $I$ we construct sequences of linear polynomials that form initially regular sequences on $R/I$. We identify situations where initially regular sequences are also regular sequences, and we show that our results can be combined with polarization to improve known depth bounds for general monomial ideals.

math.AC↗

arXiv2018

Generators of reductions of ideals in a local Noetherian ring with finite residue field

Let $(R,\mathfrak{m})$ be a local Noetherian ring with residue field $k$. While much is known about the generating sets of reductions of ideals of $R$ if $k$ is infinite, the case in which $k$ is finite is less well understood. We investigate the existence (or lack thereof) of proper reductions of an ideal of $R$ and the number of generators needed for a reduction in the case $k$ is a finite field. When $R$ is one-dimensional, we give a formula for the smallest integer $n$ for which every ideal has an $n$-generated reduction. It follows that in a one-dimensional local Noetherian ring every ideal has a principal reduction if and only if the number of maximal ideals in the normalization of the reduced quotient of $R$ is at most $|k|$. In higher dimensions, we show that for any positive integer, there exists an ideal of $R$ that does not have an $n$-generated reduction and that if $n \geq \dim R$ this ideal can be chosen to be $\mathfrak{m}$-primary. In the case where $R$ is a two-dimensional regular local ring, we construct an example of an integrally closed $\mathfrak{m}$-primary ideal that does not have a $2$-generated reduction and thus answer in the negative a question raised by Heinzer and Shannon.

math.AC↗

arXiv2017

Chudnovsky's Conjecture for very general points in $\mathbb{P}_k^{N}$

We prove a long-standing conjecture of Chudnovsky for very general and generic points in $\mathbb{P}_k^N$, where $k$ is an algebraically closed field of characteristic zero, and for any finite set of points lying on a quadric, without any assumptions on $k$. We also prove that for any homogeneous ideal $I$ in the homogeneous coordinate ring $R=k[x_0, \ldots, x_N]$, Chudnovsky's conjecture holds for large enough symbolic powers of $I$.

math.AC↗

arXiv2015

A Lower Bound For Depths of Powers of Edge Ideals

Let $G$ be a graph and let $I$ be the edge ideal of $G$. Our main results in this article provide lower bounds for the depth of the first three powers of $I$ in terms of the diameter of $G$. More precisely, we show that $\depth R/I^t \geq \left\lceil{\frac{d-4t+5}{3}} \right\rceil +p-1$, where $d$ is the diameter of $G$, $p$ is the number of connected components of $G$ and $1 \leq t \leq 3$. For general powers of edge ideals we show

math.AC↗

arXiv2013

Rees algebras of square-free monomial ideals

We study the defining equations of the Rees algebra of square-free monomial ideals in a polynomial ring over a field. We determine that when an ideal $I$ is generated by $n$ square-free monomials of the same degree then $I$ has relation type at most $n-2$ as long as $n \leq 5$. In general, we establish the defining equations of the Rees algebra in this case. Furthermore, we give a class of examples with relation type at least $n-2$, where $n=μ(I)$. We also provide new classes of ideals of linear type. We propose the construction of a graph, namely the generator graph of an ideal, where the monomial generators serve as vertices for the graph. We show that when $I$ is a square-free monomial ideal generated in the same degree that is at least 2 and the generator graph of $I$ is the graph of a disjoint union of trees and graphs with unique odd cycles then $I$ is an ideal of linear type.

math.AC↗

arXiv2012

Reduction Numbers and Balanced Ideals

Let $R$ be a Noetherian local ring and let $I$ be an ideal in $R$. The ideal $I$ is called balanced if the colon ideal $J:I$ is independent of the choice of the minimal reduction $J$ of $I$. Under suitable assumptions, Ulrich showed that $I$ is balanced if and only if the reduction number, $r(I)$, of $I$ is at most the `expected' one, namely $\ell(I)- \height I+1$, where $\ell(I)$ is the analytic spread of $I$. In this article we propose a generalization of balanced. We prove under suitable assumptions that if either $R$ is one-dimensional or the associated graded ring of $I$ is Cohen-Macaulay, then $J^{n+1}:I^n$ is independent of the choice of the minimal reduction $J$ of $I$ if and only if $r(I) \leq \ell(I)-\height I+n$.

math.AC↗