arXiv subjects

Irem Portakal

Publications and source records attributed to Irem Portakal.

At least 19 recordsLinked to original sources

arXiv2026

Stable Coexistence in Ecologies and Games

We study feasible stable equilibria of Lotka-Volterra systems and their higher-order extensions. We complete the classification of impossible ecological interaction networks with at most four species and extend several of these impossibility results to families with arbitrarily many species. We then show that these sign-pattern obstructions are specific to the pairwise Lotka-Volterra model: arbitrary prescribed growth rates and pairwise coefficients can be supplemented by higher-order interactions so as to admit a feasible asymptotically stable equilibrium. Through the correspondence with replicator dynamics, we interpret feasible equilibria of higher-order Lotka-Volterra systems as totally mixed symmetric Nash equilibria of symmetric multiplayer games, derive bounds on their number, and study their robustness under perturbations of the payoff tensors. We conclude by showing that every impossible ecology determines a nonempty open class of symmetric two-player games with no totally mixed evolutionarily stable strategy.

math.DS↗

arXiv2026

On the dimensions of correlated equilibrium polytopes of generic games

In this paper, we study the dimension of the correlated equilibrium polytope of finite games. Under the oriented-matroid notion of genericity, we prove that if a generic game is not full-dimensional, then there exists a subgame whose correlated equilibrium polytope is affinely isomorphic to that of the original game. This settles and generalizes an earlier conjecture of Brandenburg, Hollering, and Portakal (2024). Moreover, we show that the existence of a correlated equilibrium whose slices are all non-zero implies that the correlated equilibrium polytope is either full-dimensional or a singleton.

math.CO↗

arXiv2026

On the Coupled Cluster Doubles Truncation Variety of Four Electrons

We extend recent algebro-geometric results for coupled cluster theory of quantum many-body systems to the truncation varieties arising from the doubles approximation (CCD), focusing on the first genuinely nonlinear doubles regime of four electrons. Since this doubles truncation variety does not coincide with previously studied varieties, we initiate a systematic investigation of its basic algebro-geometric invariants. Combining theoretical and numerical results, we show that for $4$ electrons on $n\leq 12$ orbitals, the CCD truncation variety is a complete intersection of degree $2^{\binom{n-4}{4}}$. Using representation-theoretic arguments, we uncover a Pfaffian structure governing the quadratic relations that define the truncation variety for any $n$, and show that an exact tensor product factorization holds in a distinguished limit of disconnected doubles. We connect these structural results to the computation of the beryllium insertion into molecular hydrogen ({Be$\cdots$H$_2$ $\to$ H--Be--H}), a small but challenging bond formation process where multiconfigurational effects become pronounced.

math.AG↗

arXiv2026

A vector bundle approach to Nash equilibria

We use vector bundles to study the locus of totally mixed Nash equilibria of an $n$-player game in normal form, which we call the Nash equilibrium scheme. When the payoff tensor format is balanced, we study the Nash discriminant variety, i.e., the algebraic variety of games whose Nash equilibrium scheme is nonreduced or has a positive dimensional component. We prove that this variety has codimension one. We classify all possible components of the Nash equilibrium scheme for a binary three-player game. We prove that if the payoff tensor is of boundary format, then the Nash discriminant variety has two components: an irreducible hypersurface and a larger-codimensional component. A generic game with an unbalanced payoff tensor format does not admit totally mixed Nash equilibria. We define the Nash resultant variety of games admitting a positive number of totally mixed Nash equilibria. We prove that it is irreducible and determine its codimension and degree.

cs.GT↗

arXiv2026

Matching Criterion for Identifiability in Sparse Factor Analysis

Factor analysis models explain dependence among observed variables by a smaller number of unobserved factors. A main challenge in confirmatory factor analysis is determining whether the factor loading matrix is identifiable from the observed covariance matrix. The factor loading matrix captures the linear effects of the factors and, if unrestricted, can only be identified up to an orthogonal transformation of the factors. However, in many applications the factor loadings exhibit an interesting sparsity pattern that may lead to identifiability up to column signs. We study this phenomenon by connecting sparse confirmatory factor analysis models to bipartite graphs and providing sufficient graphical conditions for identifiability of the factor loading matrix up to column signs. In contrast to previous work, our main contribution, the matching criterion, exploits sparsity by operating locally on the graph structure, thereby improving existing conditions. Our criterion is efficiently decidable in time that is polynomial in the size of the graph, when restricting the search steps to sets of bounded size.

math.ST↗

arXiv2025

Totally mixed conditional independence equilibria of generic games

This paper further develops the algebraic--geometric foundations of conditional independence (CI) equilibria, a refinement of dependency equilibria that integrates conditional independence relations from graphical models into strategic reasoning and thereby subsumes Nash equilibria. Extending earlier work on binary games, we analyze the structure of the associated Spohn CI varieties for generic games of arbitrary format. We show that for generic games the Spohn CI variety is either empty or has codimension equal to the sum of the players' strategy dimensions minus the number of players in the parametrized undirected graphical model. When non-empty, the set of totally mixed CI equilibria forms a smooth manifold for generic games. For cluster graphical models, we introduce the class of Nash CI varieties, prove their irreducibility, and describe their defining equations, degrees, and conditions for the existence of totally mixed CI equilibria for generic games.

math.AG↗

arXiv2025

Torus Actions on Matrix Schubert and Kazhdan-Lusztig Varieties, and their Links to Statistical Models

We investigate the toric geometry of two families of generalised determinantal varieties arising from permutations: Matrix Schubert varieties ($\overline{X_w}$) and Kazhdan-Lusztig varieties ($\mathcal{N}_{v,w}$). Matrix Schubert varieties can be written as $\overline{X_w} = Y_w \times \mathbb C^d$, where $d$ is maximal. We are especially interested in the structure and complexity of these varieties $Y_w$ and $\mathcal{N}_{v,w}$ under the so-called usual torus actions. In the case when $Y_w$ is toric, we provide a full characterisation of the simple reflections $s_i$ that render ${Y_{w \cdot s_i}}$ toric, as well as the corresponding changes to the weight cone. For Kazhdan-Lusztig varieties, we consider how moving one of the two permutations $v,w$ along a chain in the Bruhat poset affects their complexity. Additionally, we study the complexity of these varieties, for permutations $v$ and $w$ of a specific structure. Finally, we consider the links between these determinantal varieties and two classes of statistical models; namely conditional independence and quasi-independence models.

math.AG↗

arXiv2025

The GameTheory package for Macaulay2

We describe the GameTheory package version 1.0 for computing equilibria in game theory available since version 1.25.05 of Macaulay2. We briefly explain the four equilibrium notions, Nash, correlated, dependency, and conditional independence, and demonstrate their implementation in the package with examples.

math.AG↗

arXiv2025

Dependency equilibria: Boundary cases and their real algebraic geometry

This paper is a significant step forward in understanding dependency equilibria within the framework of real algebraic geometry encompassing both pure and mixed equilibria. In alignment with Spohn's original definition of dependency equilibria, we propose two alternative definitions, allowing for an algebro-geometric comprehensive study of all dependency equilibria. We give a sufficient condition for the existence of a pure dependency equilibrium and show that every Nash equilibrium lies on the Spohn variety, the algebraic model for dependency equilibria. For generic games, the set of real points of the Spohn variety is Zariski dense. Furthermore, every Nash equilibrium in this case is a dependency equilibrium. Finally, we present a detailed analysis of the geometric structure of dependency equilibria for $(2\times2)$-games

math.AG↗

arXiv2025

Elliptic curves in game theory

We investigate Spohn curves, the algebro-geometric models of totally mixed dependency equilibria for $2 \times 2$ normal-form games. These curves arise as the intersection of two quadrics in $\mathbb{P}^3$ and are generically elliptic curves. We examine the reduction of Spohn curves to plane curves, providing a full classification of conditions under which they are reducible. Notably, we prove that the real points are dense on the Spohn curve in all cases, which is relevant for applications. These computations are further supported by Macaulay2 and stored in Mathrepo. We review methods to compute the $j$-invariants of elliptic curves arising as the intersection of quadrics in $\mathbb P^3$ which we apply to the case of Spohn curves aimed at game theorists. We propose a definition of equivalence of generic $2\times 2$ games based on the $j$-invariant of the Spohn curve.

math.AG↗

arXiv2024

Algebraic Sparse Factor Analysis

Factor analysis is a statistical technique that explains correlations among observed random variables with the help of a smaller number of unobserved factors. In traditional full factor analysis, each observed variable is influenced by every factor. However, many applications exhibit interesting sparsity patterns, that is, each observed variable only depends on a subset of the factors. In this paper, we study such sparse factor analysis models from an algebro-geometric perspective. Under mild conditions on the sparsity pattern, we examine the dimension of the set of covariance matrices that corresponds to a given model. Moreover, we study algebraic relations among the covariances in sparse two-factor models. In particular, we identify cases in which a Gröbner basis for these relations can be derived via a 2-delightful term order and join of toric ideals of graphs.

math.ST↗

arXiv2024

Game theory of undirected graphical models

An $n$-player game $X$ in normal form can be modeled via undirected discrete graphical models where the discrete random variables represent the players and their state spaces are the set of pure strategies. There exists an edge between the vertices of the graphical model whenever there is a dependency between the associated players. We study the Spohn conditional independence (CI) variety $\mathcal{V}_{X,\mathcal{C}}$, which is the intersection of the independence model $\mathcal{M}_{\mathcal{C}}$ with the Spohn variety of the game $X$. We prove a conjecture by the first author and Sturmfels that $\mathcal{V}_{X,\mathcal{C}}$ is of codimension $n$ in $\mathcal{M}_{\mathcal{C}}$ for a generic game $X$ with binary choices. We show that the set of totally mixed CI equilibria i.e., the restriction of the Spohn CI variety to the open probability simplex is a smooth semialgebraic manifold for a generic game $X$ with binary choices. If the undirected graph is a disjoint union of cliques, we analyze certain algebro-geometric features of Spohn CI varieties and prove affine universality theorems.

math.AG↗

arXiv2024

Combinatorics of Correlated Equilibria

We study the correlated equilibrium polytope $P_G$ of a game $G$ from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes and prove that it is a semialgebraic set for any game. Using a stratification via oriented matroids, we propose a structured method for describing the possible combinatorial types of $P_G$, and show that for $(2 \times n)$-games, the algebraic boundary of the stratification is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for generic $(2 \times 3)$-games.

math.CO↗

arXiv2023

Nash conditional independence curve

We study the Spohn conditional independence (CI) variety $C_X$ of an $n$-player game $X$ for undirected graphical models on $n$ binary random variables consisting of one edge. For a generic game, we show that $C_X$ is a smooth irreducible complete intersection curve (Nash conditional independence curve) in the Segre variety $(\mathbb{P}^{1})^{n-2} \times \mathbb{P}^3$ and we give an explicit formula for its degree and genus. We prove two universality theorems for $C_X$: The product of any affine real algebraic variety with the real line or any affine real algebraic variety in $\mathbb{R}^m$ defined by at most $m-1$ polynomials is isomorphic to an affine open subset of $C_X$ for some game $X$.

math.AG↗

arXiv2022

Complexity of the usual torus action on Kazhdan-Lusztig varieties

We investigate the class of Kazhdan-Lusztig varieties, and its subclass of matrix Schubert varieties, endowed with a naturally defined torus action. Writing a matrix Schubert variety $\overline{X_w}$ as $\overline{X_w}=Y_w\times \mathbb{C}^d$ (where $d$ is maximal possible), we show that $Y_w$ can be of complexity-$k$ exactly when $k\neq 1$. Also, we give a combinatorial description of the extremal rays of the weight cone of a Kazhdan-Lusztig variety, which in particular turns out to be the edge cone of an acyclic directed graph. As a consequence we show that given permutations $v$ and $w$, the complexity of Kazhdan-Lusztig variety indexed by $(v,w)$ is the same as the complexity of the Richardson variety indexed by $(v,w)$. Finally, we use this description to compute the complexity of certain Kazhdan-Lusztig varieties.

math.AG↗

arXiv2022

Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing

Characteristic imsets are 0-1 vectors which correspond to Markov equivalence classes of directed acyclic graphs. The study of their convex hull, named the characteristic imset polytope, has led to new and interesting geometric perspectives on the important problem of causal discovery. In this paper we begin the study of the associated toric ideal. We develop a new generalization of the toric fiber product, which we call a quasi-independence gluing, and show that under certain combinatorial homogeneity conditions, one can iteratively compute a Gröbner basis via lifting. For faces of the characteristic imset polytope associated to trees, we apply this technique to compute a Gröbner basis for the associated toric ideal. We end with a study of the characteristic ideal of the cycle and propose directions for future work.

math.ST↗

arXiv2022

Families of polytopes with rational linear precision in higher dimensions

In this article we introduce a new family of lattice polytopes with rational linear precision. For this purpose, we define a new class of discrete statistical models that we call multinomial staged tree models. We prove that these models have rational maximum likelihood estimators (MLE) and give a criterion for these models to be log-linear. Our main result is then obtained by applying Garcia-Puente and Sottile's theorem that establishes a correspondence between polytopes with rational linear precision and log-linear models with rational MLE. Throughout this article we also study the interplay between the primitive collections of the normal fan of a polytope with rational linear precision and the shape of the Horn matrix of its corresponding statistical model. Finally, we investigate lattice polytopes arising from toric multinomial staged tree models, in terms of the combinatorics of their tree representations.

math.CO↗

arXiv2022

Rigid toric matrix Schubert varieties

For a given permutation $π\in S_N$, Fulton proves that the matrix Schubert variety $\overline{X_π} \cong Y_π \times \mathbb{C}^q$ can be defined via certain rank conditions encoded in the Rothe diagram of $π$. In the case where $Y_π:=\text{TV}(σ_π)$ is toric (with respect to a $(\mathbb{C}^*)^{2N-1}$ action), we show that it can be described as an edge ideal of a bipartite graph $G^π$. We characterize the lower dimensional faces of the associated so-called edge cone $σ_π$ explicitly in terms of subgraphs of $G^π$ and present a combinatorial study for the first order deformations of $Y_π$. We prove that $Y_π$ is rigid if and only if the three-dimensional faces of $σ_π$ are all simplicial. Moreover, we reformulate this result in terms of Rothe diagram of $π$.

math.AG↗