arXiv subjects

Fatemeh Mohammadi

Publications and source records attributed to Fatemeh Mohammadi.

At least 19 recordsLinked to original sources

arXiv2026

Matroid liftability and Oxley--Wang derived matroids, with connections to scene analysis

Liftability was first introduced for paving matroids, where it was used to systematically construct polynomials in the ideals of the associated matroid varieties. In this paper, we generalise the notion of liftability to matroids with realisable truncations. We additionally introduce the notion of generic realisability of a matroid; a strong notion of realisability that implies both realisability and liftability of the matroid. We study the behaviour of liftability and generic realisability under the weak order of matroids within the lattice of matroids with fixed truncation. We develop a unified framework for studying liftability of matroids through Oxley--Wang derived matroid of generic matroid realisations. We characterise liftability and generic realisability in terms of Oxley--Wang derived matroids. Moreover, we give an algebraic description of the bases of the generic Oxley--Wang derived matroids in terms of coordinate sections of matroid realisation spaces and relate our constructions to slack realisation spaces of matroids. As an application of our framework, we obtain a combinatorial characterisation of liftable and generically realisable paving matroids and connect it with Whiteley's theory of scene analysis. In particular, our approach gives a new proof, via Oxley--Wang derived matroids, of Whiteley's characterisation of hypergraphs admitting non-trivial (sharp) scenes. Finally, we compute the generic Oxley--Wang derived matroid for a special class of sparse paving matroids.

math.CO↗

arXiv2026

A Block Tensor Train Burer-Monteiro Framework for Low-Rank Quantum State Tomography

Quantum state tomography is a fundamental technique for estimating the state of a quantum system from measured data and plays a crucial role in evaluating the performance of quantum devices. However, standard estimation methods become computationally prohibitive as the system size increases due to the exponential growth of the density matrix, describing a quantum state, with the number of qubits. We propose a low-rank tensor-network framework for mixed-state quantum state tomography based on a block tensor train (Block-TT) factorization. Specifically, the density matrix is represented as the contraction of a Block-TT with its Hermitian transpose, yielding a TT analogue of the Burer-Monteiro factorization. This parameterization guarantees Hermiticity and positive semidefiniteness by construction while compressing the number of optimization variables from exponential to linear in the number of qubits. Building on this representation, we develop single-site and two-site density matrix renormalization group (DMRG) algorithms for estimating quantum states from compressed measurements. The resulting methods operate directly on the compressed parameterization, support adaptive rank refinement, and exploit efficient tensor-network contractions for expectation-value evaluation. The framework is applicable to a broad class of low-rank quantum states, including pure states, nearly pure states, and ground states that admit accurate tensor-network approximations. Numerical experiments demonstrate accurate state reconstruction from limited measurements together with substantial reductions in memory requirements and computational cost compared with conventional low-rank tomography methods.

quant-ph↗

arXiv2026

Discrete-time, discrete-state multistate Markov models from the perspective of algebraic statistics

We study discrete-time, discrete-state multistate Markov models from the perspective of algebraic statistics. These models are widely studied in event history analysis, and are characterized by the state space, the initial distribution and the transition probabilities. A finite path under the multistate Markov model is a particular set of states occupied at finite time instances $\{1, \dots, n\}$. The main goal of this paper is to establish a bridge between event history analysis and algebraic statistics. The joint probabilities of finite paths in these models have a natural monomial parametrization in terms of the initial distribution and the transition probabilities. We study the polynomial relations among joint path probabilities. When the statistical constraints on the parameters are disregarded, nonhomogeneous multistate Markov models of arbitrary order can be viewed as slices of decomposable hierarchical models. This yields a complete description of their vanishing ideals as toric ideals generated by explicit families of binomials. Moreover, the variety of this vanishing ideal equals the nonhomogeneous multistate Markov model on the probability simplex. In contrast, homogeneous multistate Markov models exhibit different algebraic behavior, as time homogeneity imposes additional polynomial relations, leading to vanishing ideals that are strictly larger than in the nonhomogeneous case. We also derive families of binomial relations that vanish on homogeneous multistate Markov models. We investigate maximum likelihood estimation from statistical and algebraic perspectives. For nonhomogeneous models, classical and algebraic formulas agree; in the homogeneous case, the algebraic approach is more complex. Lastly, we provide data applications where we demonstrate the statistical theory to obtain the maximum likelihood estimates of the parameters under specific multistate Markov models.

math.ST↗

arXiv2026

Terracini matroids: algebraic matroids of secants and embedded joins

Applications of algebraic geometry have sparked much recent work on algebraic matroids. An algebraic matroid encodes algebraic dependencies among coordinate functions on a variety. We study the behavior of algebraic matroids under joins and secants of varieties. Motivated by Terracini's lemma, we introduce the notion of a Terracini union of matroids, which captures when the algebraic matroid of a join coincides with the matroid union of the algebraic matroids of its summands. We illustrate applications of our results with a discussion of the implications for toric surfaces and threefolds.

math.CO↗

arXiv2026

Asymptotic properties of random monomial ideals

This paper focuses on asymptotic properties of random monomial ideals through a statistical viewpoint. It extends the study of redundancy in monomial ideals by analyzing the poset density of the LCM-lattice. We explore how this density behaves across random algebraic models and structured networks. Experimental data reveal that the LCM-lattice exhibits sharp threshold behavior rather than changing smoothly. We observe a strong negative correlation between the number of generators and LCM-lattice density, abruptly separating three distinct regimes: a low-density Taylor-like regime, a high-density redundant regime, and a narrow transition window. We show that increasing the generator degree causes this density drop to occur at lower probability thresholds. We conclude by conjecturing that for equigenerated squarefree ideals, the LCM-lattice density undergoes a sharp phase transition, analogous to the emergence of giant components in hypergraphs. This suggests that the classical, ideal-by-ideal role of the LCM-lattice as a combinatorial invariant also admits a statistical/asymptotic counterpart: in natural random families, redundancy and resolution-complexity indicators concentrate into distinct typical regimes separated by a narrow transition window.

cs.SC↗

arXiv2026

Gröbner bases, resolutions, and the Lefschetz properties for powers of a general linear form in the squarefree algebra

For the almost complete intersection ideals $(x_1^2, \dots, x_n^2, (x_1 + \cdots + x_n)^k)$, we compute their reduced Gröbner basis for any term ordering, revealing a combinatorial structure linked to lattice paths, elementary symmetric polynomials, and Catalan numbers. Using this structure, we classify the weak Lefschetz property for these ideals. Additionally, we provide a new proof of the well-known result that the squarefree algebra satisfies the strong Lefschetz property. Finally, we compute the Betti numbers of the initial ideals and construct a minimal free resolution using a Mayer-Vietoris tree approach.

math.AC↗

arXiv2026

Paving Matroids: Defining Equations and Associated Varieties

We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of these varieties across any dimension. Additionally, we explain the relationship between polynomials constructed using these different methods. We then compute a comprehensive and finite set of defining equations for matroid varieties associated with specific classes of paving matroids. Finally, we focus on the class of paving matroids of rank $3$, known as point-line configurations, which essentially contain simple matroids of rank $3$. Furthermore, we provide a decomposition for the associated circuit variety of point-line configurations, where all points have a degree less than $3$. Lastly, we present several examples applying our results and compare them with the known cases in the literature.

math.AG↗

arXiv2026

Decomposing Determinantal Varieties from Statistics via Matroid Theory

We study determinantal varieties from conditional independence models with hidden variables, focusing on their irreducible decompositions, dimensions, degrees, and Gröbner bases. Each variety encodes a collection of matroids, whose flats capture algebraic dependencies among variables. Using this approach, we provide a systematic description of the components, their dimensions, and defining equations, and introduce a combinatorial framework for computing the degree of the determinantal variety. Our approach highlights the central role of matroidal structures in the study of determinantal varieties and extends beyond the reach of current computational techniques.

math.CO↗

arXiv2026

Algebraic and Algorithmic Methods for Computing Polynomial Loop Invariants

Loop invariants are properties of a program loop that hold both before and after each iteration of the loop. They are often used to verify programs and ensure that algorithms consistently produce correct results during execution. Consequently, generating invariants becomes a crucial task for loops. We specifically focus on polynomial loops, where both the loop conditions and the 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 can have arbitrary degrees. Using tools from algebraic geometry, we present two algorithms designed to generate all polynomial invariants within a given vector subspace, for a branching loop with nondeterministic conditional statements. These algorithms combine linear algebraic subroutines with computations on polynomial ideals. They differ depending on whether the initial values of the loop variables are specified or treated as parameters. Additionally, we present a much more efficient algorithm for generating polynomial invariants of a specific form, applicable to all initial values. This algorithm avoids expensive ideal computations.

cs.SC↗

arXiv2025

Single-cell 3D genome reconstruction in the haploid setting using rigidity theory

This article considers the problem of 3-dimensional genome reconstruction for single-cell data, and the uniqueness of such reconstructions in the setting of haploid organisms. We consider multiple graph models as representations of this problem, and use techniques from graph rigidity theory to determine identifiability. Biologically, our models come from Hi-C data, microscopy data, and combinations thereof. Mathematically, we use unit ball and sphere packing models, as well as models consisting of distance and inequality constraints. In each setting, we describe and/or derive new results on realisability and uniqueness. We then propose a 3D reconstruction method based on semidefinite programming and apply it to synthetic and real data sets using our models.

q-bio.GN↗

arXiv2025

Phase diagram of the Kitaev-Hubbard model: $\mathbb{Z}_2$ slave-spin and QMC approaches

Recent experiments show that the ground state of some layered materials with localized moments is in close proximity to the Kitaev spin liquid, calling for a proper model to describe the measurements. The Kitaev-Hubbard (KHu) model is the minimal model that captures the essential ingredients of these systems; it yields the Kitaev-Heisenberg spin model at the strong coupling limit and contains the charge fluctuations present in these materials as well. Despite its relevance, the phase diagram of the KHu model has not been rigorously revealed yet. In this work, we study the full phase diagram of the KHu model using the $\mathbb{Z}_2$ slave-spin mean-field theory as well as the auxiliary field quantum Monte Carlo on rather large systems and at low temperatures. The Mott transition is signaled by a vanishing quasiparticle weight evaluated using the slave-spin construction. Moreover, we demonstrate that there are multiple magnetic phase transitions within the Mott phase including magnetically ordered phases and most notably a quantum spin liquid phase for $1.0 \lesssim t^{\prime}/t \lesssim 1.11 $ at $U/t=5$.

cond-mat.str-el↗

arXiv2025

Decomposing conditional independence ideals with hidden variables

We study a family of determinantal ideals whose decompositions encode the structural zeros in conditional independence models with hidden variables. We provide explicit decompositions of these ideals and, for certain subclasses of models, we show that this is a decomposition into radical ideals by displaying Gröbner bases for the components. We identify conditions under which the components are prime, and establish formulas for the dimensions of these prime ideals. Moreover, we show that the components in the decomposition can be grouped into equivalence classes defined by their combinatorial structure, and we derive a closed formula for the number of such classes.

math.AC↗

arXiv2025

Efficient Algorithms for Maximal Matroid Degenerations and Irreducible Decompositions of Circuit Varieties

Matroid theory provides a unifying framework for studying dependence across combinatorics, geometry, and applications ranging from rigidity to statistics. In this work, we study circuit varieties of matroids, defined by their minimal dependencies, which play a central role in modeling determinantal varieties, rigidity problems, and conditional independence relations. We introduce an efficient computational strategy for decomposing the circuit variety of a given matroid $M$, based on an algorithm that identifies its maximal degenerations. These degenerations correspond to the largest matroids lying below $M$ in the weak order. Our framework yields explicit and computable decompositions of circuit varieties that were previously out of reach for symbolic or numerical algebra systems. We apply our strategy to several classical configurations, including the Vámos matroid, the unique Steiner quadruple system $S(3,4,8)$, projective and affine planes, the dual of the Fano matroid, and the dual of the graphic matroid of $K_{3,3}$. In each case, we successfully compute the minimal irreducible decomposition of their circuit varieties.

math.CO↗

arXiv2025

Solvable and Nilpotent Matroids: Realizability and Irreducible Decomposition of Their Associated Varieties

We introduce the families of solvable and nilpotent matroids, examining their realization spaces, closures, and associated matroid and circuit varieties. We study their realizability, as well as the irreducible decomposition of their associated matroid and circuit varieties. Additionally, we describe a finite generating set for the corresponding ideals, considered up to radical. We establish sufficient conditions for both the realizability of these matroids and the irreducibility of their associated varieties. Specifically, we establish the realizability and irreducibility of matroid varieties associated with nilpotent matroids and prove the irreducibility of matroid varieties arising from certain classes of solvable paving matroids. Additionally, we analyze the defining polynomial equations of these varieties using Grassmann-Cayley algebra and geometric liftability techniques. Furthermore, we provide a complete generating set for the matroid ideals associated with forest configurations.

math.CO↗

arXiv2025

Vandermonde Cells Through the Lens of Positive Geometry

We study the geometric and algebraic structure of Vandermonde cells, defined as images of the standard probability simplex under the Vandermonde map given by consecutive power sum polynomials. Motivated by their combinatorial equivalence to cyclic polytopes, which are well-known examples of positive geometries and tree amplituhedra, we investigate whether Vandermonde cells admit the structure of positive geometries. We derive explicit parametrizations and algebraic equations for their boundary components, extending known results from the planar case to arbitrary dimensions. By introducing a mild generalization of the notion of positive geometry, allowing singularities within boundary interiors, we show that planar Vandermonde cells naturally fit into this extended framework. Furthermore, we study Vandermonde cells in the setting of Brown-Dupont's mixed Hodge theory formulation of positive geometries, and show that they form a genus zero pair. These results provide a new algebraic and geometric understanding of Vandermonde cells, establishing them as promising examples within the emerging theory of positive geometries.

math.CO↗

arXiv2025

From Affine to Polynomial: Synthesizing Loops with Branches via Algebraic Geometry

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 such invariants is a crucial task in loop analysis, but it is undecidable 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. We use algebraic geometry tools to design and implement an algorithm that computes a finite set of polynomial equations whose solutions correspond to all nondeterministic branching loops satisfying the given invariants. Furthermore, we introduce a new class of invariants for which we present a significantly more efficient algorithm. In other words, we reduce the problem of synthesizing loops to find solutions of multivariate polynomial systems with rational entries. This final step is handled in our software using an SMT solver.

cs.PL↗

arXiv2025

Characterizing Translation-Invariant Bell Inequalities using Tropical Algebra and Graph Polytopes

Nonlocality is one of the key features of quantum physics, which is revealed through the violation of a Bell inequality. In large multipartite systems, nonlocality characterization quickly becomes a challenging task. A common practice is to make use of symmetries, low-order correlators, or exploiting local geometries, to restrict the class of inequalities. In this paper, we characterize translation-invariant (TI) Bell inequalities with finite-range correlators in one-dimensional geometries. We introduce a novel methodology based on tropical algebra tensor networks and highlight its connection to graph theory. Surprisingly, we find that the TI Bell polytope has a number of extremal points that can be uniformly upper-bounded with respect to the system size. We give an efficient method to list all vertices of the polytope for a particular system size, and characterize the tightness of a given TI Bell inequality. The connections highlighted in our work allow us to re-interpret concepts developed in the fields of tropical algebra and graph theory in the context of Bell nonlocality, and vice-versa. This work extends a parallel article [M. Hu \textit{et al.}, arXiv: 2208.02798 (2022)] on the same subject.

quant-ph↗

arXiv2025

Computing positroid cells in the Grassmannian of lines, their boundaries and their intersections

Positroids are families of matroids introduced by Postnikov in the study of non-negative Grassmannians. In particular, positroids enumerate a CW decomposition of the totally non-negative Grassmannian. Furthermore, Postnikov has identified several families of combinatorial objects in bijections with positroids. We will provide yet another characterization of positroids for Gr$_{\geq 0}(2,n)$, the Grassmannians of lines, in terms of certain graphs. We use this characterization to compute the dimension and the boundary of positroid cells. This also leads to a combinatorial description of the intersection of positroid cells, that is easily computable. Our techniques rely on determining different ways to enlarge a given collection of subsets of $\{1,\ldots,n\}$ to represent the dependent sets of a positroid, that is the dependencies among the columns of a matrix with non-negative maximal minors. Furthermore, we provide an algorithm to compute all the maximal positroids contained in a set.

math.CO↗