arXiv subjects

Fatemeh Mohammadi

Publications and source records attributed to Fatemeh Mohammadi.

At least 37 records · Page 2Linked to original sources

arXiv2025

The Gröbner basis for powers of a general linear form in a monomial complete intersection

We study almost complete intersection ideals in a polynomial ring, generated by powers of all the variables together with a power of their sum. Our main result is an explicit description of the reduced Gröbner bases for these ideals under any term order. Our approach is primarily combinatorial, focusing on the structure of the initial ideal. We associate a lattice path to each monomial in the vector space basis of an Artinian monomial complete intersection and introduce a reflection operation on these paths, which enables a key counting argument. As a consequence, we provide a new proof that Artinian monomial complete intersections possess the strong Lefschetz property over fields of characteristic zero. Our results also offer new insights into the longstanding problem of classifying the weak Lefschetz property for such intersections in characteristic $p$. Furthermore, we show that the number of Gröbner basis elements in each degree is connected to several well-known sequences, including the (generalized) Catalan, Motzkin, and Riordan numbers, and connect these numbers to the study of entanglement detection in spin systems within quantum physics.

math.AC↗

arXiv2025

K-polynomials of multiplicity-free varieties

We describe the twisted $K$-polynomial of multiplicity-free varieties in a multiprojective setting. More precisely, for multiplicity-free varieties, we show that the support of the twisted $K$-polynomial is a generalized polymatroid. As applications, we show that the support of the Möbius function of a linear polymatroid is a generalized polymatroid, and we settle a conjecture of Monical, Tokcan and Yong regarding Grothendieck polynomials for the case of zero-one Schubert polynomials.

math.AG↗

arXiv2025

Algebraic Geometry of Cactus, Pascal, and Pappus Matroids

We study rank-three matroids, known as point-line configurations, and their associated matroid varieties, defined as the Zariski closures of their realization spaces. Our focus is on determining finite generating sets of defining equations for these varieties, up to radical, and describing the irreducible components of the corresponding circuit varieties. We generalize the notion of cactus graphs to matroids, introducing a family of point-line configurations whose underlying graphs are cacti. Our analysis includes several classical matroids, such as the Pascal, Pappus, and cactus matroids, for which we provide explicit finite generating sets for their associated matroid ideals. The matroid ideal is the ideal of the matroid variety, whose construction involves a saturation step with respect to all the independence relations of the matroid. This step is computationally very expensive and has only been carried out for very small matroids. We provide a complete generating set of these ideals for the Pascal, Pappus, and cactus matroids. The proofs rely on classical geometric techniques, including liftability arguments and the Grassmann--Cayley algebra, which we use to construct so-called bracket polynomials in these ideals. In addition, we prove that every cactus matroid is realizable and that its matroid variety is irreducible.

math.CO↗

arXiv2025

Redundancy analysis using lcm-filtrations: networks, system signature and sensitivity evaluation

We introduce the lcm-filtration and stepwise filtration, comparing their performance across various scenarios in terms of computational complexity, efficiency, and redundancy. The lcm-filtration often involves identical steps or ideals, leading to unnecessary computations. To address this, we analyse how stepwise filtration can effectively compute only the non-identical steps, offering a more efficient approach. We compare these filtrations in applications to networks, system signatures, and sensitivity analysis.

cs.DM↗

arXiv2025

Beyond Affine Loops: A Geometric Approach to Program Synthesis

Ensuring software correctness remains a fundamental challenge in formal program verification. One promising approach relies on finding polynomial invariants for loops. Polynomial invariants are properties of a program loop that hold before and after each iteration. Generating polynomial invariants is a crucial task for loops, but it is an undecidable problem in the general case. Recently, an alternative approach to this problem has emerged, focusing on synthesizing loops from invariants. However, existing methods only synthesize affine loops without guard conditions from polynomial invariants. In this paper, we address a more general problem, allowing loops to have polynomial update maps with a given structure, inequations in the guard condition, and polynomial invariants of arbitrary form. In this paper, we use algebraic geometry tools to design and implement an algorithm that computes a finite set of polynomial equations whose solutions correspond to all loops satisfying the given polynomial invariants. In other words, we reduce the problem of synthesizing loops to finding solutions of polynomial systems within a specified subset of the complex numbers. The latter is handled in our software using an SMT solver.

cs.SC↗

arXiv2025

Artificial Conversations, Real Results: Fostering Language Detection with Synthetic Data

Collecting high-quality training data is essential for fine-tuning Large Language Models (LLMs). However, acquiring such data is often costly and time-consuming, especially for non-English languages such as Italian. Recently, researchers have begun to explore the use of LLMs to generate synthetic datasets as a viable alternative. This study proposes a pipeline for generating synthetic data and a comprehensive approach for investigating the factors that influence the validity of synthetic data generated by LLMs by examining how model performance is affected by metrics such as prompt strategy, text length and target position in a specific task, i.e. inclusive language detection in Italian job advertisements. Our results show that, in most cases and across different metrics, the fine-tuned models trained on synthetic data consistently outperformed other models on both real and synthetic test datasets. The study discusses the practical implications and limitations of using synthetic data for language detection tasks with LLMs.

cs.CL↗

arXiv2025

Identifying Gender Stereotypes and Biases in Automated Translation from English to Italian using Similarity Networks

This paper is a collaborative effort between Linguistics, Law, and Computer Science to evaluate stereotypes and biases in automated translation systems. We advocate gender-neutral translation as a means to promote gender inclusion and improve the objectivity of machine translation. Our approach focuses on identifying gender bias in English-to-Italian translations. First, we define gender bias following human rights law and linguistics literature. Then we proceed by identifying gender-specific terms such as she/lei and he/lui as key elements. We then evaluate the cosine similarity between these target terms and others in the dataset to reveal the model's perception of semantic relations. Using numerical features, we effectively evaluate the intensity and direction of the bias. Our findings provide tangible insights for developing and training gender-neutral translation algorithms.

cs.CL↗

arXiv2025

Leveraging GPT-4o Efficiency for Detecting Rework Anomaly in Business Processes

This paper investigates the effectiveness of GPT-4o-2024-08-06, one of the Large Language Models (LLM) from OpenAI, in detecting business process anomalies, with a focus on rework anomalies. In our study, we developed a GPT-4o-based tool capable of transforming event logs into a structured format and identifying reworked activities within business event logs. The analysis was performed on a synthetic dataset designed to contain rework anomalies but free of loops. To evaluate the anomaly detection capabilities of GPT 4o-2024-08-06, we used three prompting techniques: zero-shot, one-shot, and few-shot. These techniques were tested on different anomaly distributions, namely normal, uniform, and exponential, to identify the most effective approach for each case. The results demonstrate the strong performance of GPT-4o-2024-08-06. On our dataset, the model achieved 96.14% accuracy with one-shot prompting for the normal distribution, 97.94% accuracy with few-shot prompting for the uniform distribution, and 74.21% accuracy with few-shot prompting for the exponential distribution. These results highlight the model's potential as a reliable tool for detecting rework anomalies in event logs and how anomaly distribution and prompting strategy influence the model's performance.

cs.LG↗

arXiv2025

Minimal matroids in dependency posets: algorithms and applications to computing irreducible decompositions of circuit varieties

We study point-line configurations, their minimal matroids, and their associated circuit varieties. We present an algorithm for identifying the minimal matroids of these configurations with respect to dependency order, or equivalently, the maximal matroids with respect to weak order, and use it to determine the irreducible decomposition of their corresponding circuit varieties. Our algorithm is applied to several classical configurations, including the Fano matroid, affine plane of order three, MacLane, and Pappus configurations. Additionally, we explore the connection to a conjecture by Jackson and Tanigawa, which provides a criterion for the uniqueness of the minimal matroids.

math.CO↗

arXiv2025

Minimal cellular resolutions of powers of matching field ideals

We study a family of monomial ideals, called block diagonal matching field ideals, which arise as monomial Gröbner degenerations of determinantal ideals. Our focus is on the minimal free resolutions of these ideals and all of their powers. Initially, we establish their linear quotient property and compute their Betti numbers, illustrating that their minimal free resolution is supported on a regular CW complex. Our proof relies on the results of Herzog and Takayama, demonstrating that ideals with a linear quotient property have a minimal free resolution, and on the construction by Dochtermann and Mohammadi of cellular realizations of these resolutions. We begin by proving the linear quotient property for each power of such an ideal. Subsequently, we show that their corresponding decomposition map is regular, resulting in a minimal cellular resolution. Finally, we demonstrate that distinct decomposition maps lead to different cellular complexes with the same face numbers.

math.AC↗

arXiv2024

Irreducibility, Smoothness, and Connectivity of Realization Spaces of Matroids and Hyperplane Arrangements

We study the realization spaces of matroids and hyperplane arrangements. First, we define the notion of naive dimension for the realization space of matroids and compare it with the expected dimension and the algebraic dimension, exploring the conditions under which these dimensions coincide. Next, we introduce the family of inductively connected matroids and investigate their realization spaces, establishing that they are smooth, irreducible, and isomorphic to a Zariski open subset of a complex space with a known dimension. Furthermore, we present an explicit procedure for computing their defining equations. As corollaries, we identify families of hyperplane arrangements whose moduli spaces are connected. Finally, we apply our results to study the rigidity of matroids. Rigidity, which involves matroids with a unique realization under projective transformations, is key to understanding the connectivity of the moduli spaces of the corresponding hyperplane arrangements.

math.CO↗

arXiv2024

Klein-Arnold tensegrities

In this paper, we introduce new classes of infinite and combinatorially periodic tensegrities, derived from algebraic multidimensional continued fractions in the sense of F. Klein. We describe the stress coefficients on edges through integer invariants of these continued fractions, as initiated by V.I. Arnold, thereby creating a novel connection between geometric rigidity theory and the geometry of continued fractions. Remarkably, the new classes of tensegrities possess rational self-stress coefficients. To establish the self-stressability of the frameworks, we present a projective version of the classical Maxwell-Cremona lifting principle, a result of independent interest.

math.CO↗

arXiv2024

Combinatorics of Essential Sets for Positroids

Positroids are a family of matroids introduced by Postnikov in the study of non-negative Grassmannians. Postnikov identified several combinatorial objects in bijections with positroids, among which are bounded affine permutations. On the other hand, the notion of essential sets, introduced for permutations by Fulton, was used by Knutson in the study of the special family of interval rank positroids. We generalize Fulton's essential sets to bounded affine permutations. The bijection of the latter with positroids, allows study of the relationship between them. From the point of view of positroids, essential sets are maximally dependent cyclic interval. We define connected essential sets and prove that they give a facet description of the positroid polytope, as well as equations defining the positroid variety. We define a subset of essential sets, called core, which contains minimal rank conditions to uniquely recover a positroid. We provide an algorithm to retrieve the positroid satisfying the rank conditions in the core or any compatible rank condition on cyclic intervals.

math.CO↗

arXiv2024

Stable Coherent Systems

We describe the notion of stability of coherent systems as a framework to deal with redundancy. We define stable coherent systems and show how this notion can help the design of reliable systems. We demonstrate that the reliability of stable systems can be efficiently computed using the algebraic versions of improved inclusion-exclusion formulas and sum of disjoint products.

math.AC↗

arXiv2024

Algebraic Tools for Computing Polynomial Loop Invariants

Loop invariants are properties of a program loop that hold before and after each iteration of the loop. They are often employed to verify programs and ensure that algorithms consistently produce correct results during execution. Consequently, the generation of invariants becomes a crucial task for loops. We specifically focus on polynomial loops, where both the loop conditions and assignments within the loop are expressed as polynomials. Although computing polynomial invariants for general loops is undecidable, efficient algorithms have been developed for certain classes of loops. For instance, when all assignments within a while loop involve linear polynomials, the loop becomes solvable. In this work, we study the more general case where the polynomials exhibit arbitrary degrees. Applying tools from algebraic geometry, we present two algorithms designed to generate all polynomial invariants for a while loop, up to a specified degree. These algorithms differ based on whether the initial values of the loop variables are given or treated as parameters. Furthermore, we introduce various methods to address cases where the algebraic problem exceeds the computational capabilities of our methods. In such instances, we identify alternative approaches to generate specific polynomial invariants.

cs.SC↗

arXiv2024

A Family of Monomial Gröbner Degenerations of Determinantal Ideals with Minimal Cellular Resolutions

We explore a family of monomial ideals derived as Gröbner degenerations of determinantal ideals. These ideals, previously examined as block diagonal matching field ideals within the realm of toric degenerations of Grassmannians, are identified as monomial initial ideals of determinantal ideals. We establish their linear quotient property and calculate their Betti numbers, demonstrating that their minimal free resolution is supported on a regular CW complex, equivalently, they have a cellular resolution. On the combinatorial front, we examine a range of weight vectors that yield diverse monomial initial ideals for determinantal ideals. We show that despite variations in weight vectors, all these ideals share identical Betti numbers. However, altering a weight vector results in distinct cellular complexes supporting their minimal free resolutions.

math.AC↗

arXiv2024

A general formulation of reweighted least squares fitting

We present a generalized formulation for reweighted least squares approximations. The goal of this article is twofold: firstly, to prove that the solution of such problem can be expressed as a convex combination of certain interpolants when the solution is sought in any finite-dimensional vector space; secondly, to provide a general strategy to iteratively update the weights according to the approximation error and apply it to the spline fitting problem. In the experiments, we provide numerical examples for the case of polynomials and splines spaces. Subsequently, we evaluate the performance of our fitting scheme for spline curve and surface approximation, including adaptive spline constructions.

math.NA↗

arXiv2024

Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties

We study point-line configurations through the lens of projective geometry and matroid theory. Our focus is on their realisation spaces, where we introduce the concepts of liftable and quasi-liftable configurations, exploring cases in which an $n$-tuple of collinear points can be lifted to a non-degenerate realisation of a point-line configuration. We show that forest configurations are liftable and characterise the realisation space of liftable configurations as the solution set of certain linear systems of equations. Moreover, we study the Zariski closure of the realisation spaces of liftable and quasi-liftable configurations, known as matroid varieties, and establish their irreducibility. Additionally, we compute an irreducible decomposition for their corresponding circuit varieties. Applying these liftability properties, we present a procedure generate some of the defining equations of the associated matroid varieties. As corollaries, we provide a geometric representation for the defining equations of two specific examples: the quadrilateral set and the $3\times4$ grid. While the polynomials for the latter were previously computed using specialised algorithms tailored for this configuration, the geometric interpretation of these generators was missing. We compute a minimal generating set for the corresponding ideals.

math.CO↗