arXiv subjects

Luca Sodomaco

Publications and source records attributed to Luca Sodomaco.

At least 19 recordsLinked to original sources

arXiv2026

Nash Loci

In the study of Nash equilibria of finite-player games, one often seeks equilibria that are compatible with predetermined constraints, either determined by the players or by an external agent. We discuss the algebraic loci, called Nash loci, of games whose Nash equilibrium scheme intersects a fixed algebraic variety in a product of projective spaces. We determine their dimensions and multidegrees in multiprojective space, and their equations for two-player games and for small multiple-player games. The multilinear equations defining the Nash equilibrium scheme allow us to describe Nash loci in the language of Grassmannians and Pl\"{u}cker coordinates. Motivated by this fact, we relate Nash loci to multigraded associated varieties, which are subvarieties in products of Grassmannians that generalize the multigraded Cayley-Chow hypersurfaces of Osserman and Trager.

math.AG↗

arXiv2026

Benchmarks in Leipzig

Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the 3-day workshop *Benchmarks in Leipzig* with 35 participants at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. We present the resulting collection of 100 questions. We evaluated these questions in three stages: a single attempt by five state-of-the-art LLMs, followed by a 20-runs-per-model evaluation with three of these models, and finally a 3-run attempt with two heavy-thinking models. After Stage 1, 41 questions remained completely unsolved; after Stage 2, this count dropped to 16; and we concluded Stage 3 with only 2 unsolved questions. This demonstrates that the mathematical reasoning capabilities of LLMs are becoming impressive.

math.HO↗

arXiv2026

Multigraded Hurwitz forms

The Hurwitz form of a projective variety characterizes linear spaces of complementary dimension which meet the variety non-transversally. We extend this notion to varieties in a product of projective spaces. This parallels the multigraded Chow forms due to Osserman and Trager. We study the degrees of multigraded Hurwitz forms. An explicit degree formula is given for complete intersections. This offers a new tool for elimination theory that has many applications, ranging from Nash equilibria to Feynman integrals.

math.AG↗

arXiv2025

Nonlinear Kalman varieties

We study the locus of square matrices having at least one eigenvector on a prescribed algebraic variety $X$. When $X$ is a linear subspace, this data locus is known as the Kalman variety of $X$ and was studied first by Ottaviani and Sturmfels. Motivated by recent applications to quantum chemistry and optimization, in this work, we focus on nonlinear Kalman varieties, that is, Kalman varieties relative to arbitrary projective varieties $X$. We study the basic invariants of these varieties, such as their dimensions, degrees, and singularities. Furthermore, Ottaviani and Sturmfels provide determinantal equations in the linear case. We generalize their result to Kalman varieties of hypersurfaces by providing a determinantal-like description of their equation.

math.AG↗

arXiv2025

Nonlinear Rayleigh quotient optimization

Rayleigh quotient minimization deals with optimizing a quadratic homogeneous function over a sphere. Its critical points correspond to the normalized eigenvectors of the symmetric matrix associated with the quadratic form. In this paper, we consider a homogeneous polynomial objective function $f$ over a sphere, a projective algebraic variety $X$, and we study the $X$-eigenpoints of $f$, which are classes of critical points of $f$ constrained to the sphere and the affine cone over $X$. The number of $X$-eigenpoints of a generic polynomial $f$ is the Rayleigh-Ritz degree of $X$. This invariant is a version of the Euclidean distance degree of a Veronese embedding of $X$. We provide concrete formulas in various scenarios, including those involving varieties of rank-one tensors.

math.AG↗

arXiv2025

Osculating Geometry and Higher-Order Distance Loci

We discuss the problem of optimizing the distance function from a given point, subject to polynomial constraints. A key algebraic invariant that governs its complexity is the Euclidean distance degree, which pertains to first-order tangency. We focus on the data locus of points possessing at least one critical point of the distance function that is normal to a higher-order osculating space. We study the higher-order distance degree of a morphism as an intersection-theoretic invariant involving jet bundles and higher-order polar loci. Our approach builds on foundational definitions and results developed by Piene, particularly regarding higher-order polar loci. We give closed formulas for generic maps, Veronese embeddings, and toric embeddings. We place particular emphasis on the Bombieri-Weyl metric, revealing that the chosen metric profoundly influences both the degree and birationality of the higher-order projection maps. Additionally, we introduce a tropical framework that represents these degrees as stable intersections with Bergman fans, facilitating effective combinatorial computation in toric settings.

math.AG↗

arXiv2025

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↗

arXiv2023

Conditional Euclidean distance optimization via relative tangency

We introduce a theory of relative tangency for projective algebraic varieties. The dual variety $X_Z^\vee$ of a variety $X$ relative to a subvariety $Z$ is the set of hyperplanes tangent to $X$ at a point of $Z$. We also introduce the concept of polar classes of $X$ relative to $Z$. We explore the duality of varieties of low rank matrices relative to special linear sections. In this framework, we study the critical points of the Euclidean Distance function from a data point to $X$, lying on $Z$. The locus where the number of such conditional critical points is positive is called the ED data locus of $X$ given $Z$. The generic number of such critical points defines the conditional ED degree of $X$ given $Z$. We show the irreducibility of ED data loci, and we compute their dimensions and degrees in terms of relative characteristic classes.

math.AG↗

arXiv2023

On the minimal algebraic complexity of the rank-one approximation problem for general inner products

We study the algebraic complexity of Euclidean distance minimization from a generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED) degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products. We prove that Frobenius inner product is a local minimum of the ED degree, and conjecture that it is a global minimum. We prove our conjecture in the case of matrices and symmetric binary and $3\times 3\times 3$ tensors. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry.

math.AG↗

arXiv2023

The span of singular tuples of a tensor beyond the boundary format

A singular $k$-tuple of a tensor $T$ of format $(n_1,\ldots,n_k)$ is essentially a complex critical point of the distance function from $T$ constrained to the cone of tensors of format $(n_1,\ldots,n_k)$ of rank at most one. A generic tensor has finitely many complex singular $k$-tuples, and their number depends only on the tensor format. Furthermore, if we fix the first $k-1$ dimensions $n_i$, then the number of singular $k$-tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable $n_k$, that stabilizes when $(n_1,\ldots,n_k)$ reaches a boundary format. In this paper, we study the linear span of singular $k$-tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order-$k$ tensors of format $(2,\ldots,2,n)$. As a consequence, if again we fix the first $k-1$ dimensions $n_i$ and let $n_k$ increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with $k>3$. Finally, we provide equations for the linear span of singular triples of a generic order three tensor $T$ of some special non-sub-boundary format. From these equations, we conclude that $T$ belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format.

math.AG↗

arXiv2022

Dimensions of Higher Order Factor Analysis Models

The factor analysis model is a statistical model where a certain number of hidden random variables, called factors, affect linearly the behaviour of another set of observed random variables, with additional random noise. The main assumption of the model is that the factors and the noise are Gaussian random variables. This implies that the feasible set lies in the cone of positive semidefinite matrices. In this paper, we do not assume that the factors and the noise are Gaussian, hence the higher order moment and cumulant tensors of the observed variables are generally nonzero. This motivates the notion of kth-order factor analysis model, that is the family of all random vectors in a factor analysis model where the factors and the noise have finite and possibly nonzero moment and cumulant tensors up to order k. This subset may be described as the image of a polynomial map onto a Cartesian product of symmetric tensor spaces. Our goal is to compute its dimension and we provide conditions under which the image has positive codimension.

math.ST↗

arXiv2022

Log-concave density estimation in undirected graphical models

We study the problem of maximum likelihood estimation of densities that are log-concave and lie in the graphical model corresponding to a given undirected graph $G$. We show that the maximum likelihood estimate (MLE) is the product of the exponentials of several tent functions, one for each maximal clique of $G$. While the set of log-concave densities in a graphical model is infinite-dimensional, our results imply that the MLE can be found by solving a finite-dimensional convex optimization problem. We provide an implementation and a few examples. Furthermore, we show that the MLE exists and is unique with probability 1 as long as the number of sample points is larger than the size of the largest clique of $G$ when $G$ is chordal. We show that the MLE is consistent when the graph $G$ is a disjoint union of cliques. Finally, we discuss the conditions under which a log-concave density in the graphical model of $G$ has a log-concave factorization according to $G$.

math.ST↗

arXiv2022

Degrees of Kalman varieties of tensors

Kalman varieties of tensors are algebraic varieties consisting of tensors whose singular vector $k$-tuples lay on prescribed subvarieties. They were first studied by Ottaviani and Sturmfels in the context of matrices. We extend recent results of Ottaviani and the first author to the partially symmetric setting. We describe a generating function whose coefficients are the degrees of these varieties and we analyze its asymptotics, providing analytic results à la Zeilberger and Pantone. We emphasize the special role of isotropic vectors in the spectral theory of tensors and describe the totally isotropic Kalman variety as a dual variety.

math.AG↗

arXiv2022

Exact solutions in low-rank approximation with zeros

Low-rank approximation with zeros aims to find a matrix of fixed rank and with a fixed zero pattern that minimizes the Euclidean distance to a given data matrix. We study the critical points of this optimization problem using algebraic tools. In particular, we describe special linear, affine, and determinantal relations satisfied by the critical points. We also investigate the number of critical points and how this number is related to the complexity of nonnegative matrix factorization problem.

math.AG↗

arXiv2021

Asymptotics of degrees and ED degrees of Segre products

Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and their dual varieties. We analyze the asymptotics of degrees of (hypercubical) hyperdeterminants, the dual hypersurfaces to Segre varieties. We offer an alternative viewpoint on the stabilization of the ED degree of some Segre varieties. Although this phenomenon was incidentally known from Friedland-Ottaviani's formula expressing the number of singular vector tuples of a general tensor, our approach provides a geometric explanation. Finally, we establish the stabilization of the degree of the dual variety of a Segre product $X\times Q_{n}$, where $X$ is a projective variety and $Q_n\subset \mathbb{P}^{n+1}$ is a smooth quadric hypersurface.

math.AG↗

arXiv2021

Algebraic degree of optimization over a variety with an application to $p$-norm distance degree

We study an optimization problem with the feasible set being a real algebraic variety $X$ and whose parametric objective function $f_u$ is gradient-solvable with respect to the parametric data $u$. This class of problems includes Euclidean distance optimization as well as maximum likelihood optimization. For these particular optimization problems, a prominent role is played by the ED and ML correspondence, respectively. To our generalized optimization problem we attach an optimization correspondence and show that it is equidimensional. This leads to the notion of algebraic degree of optimization on $X$. We apply these results to $p$-norm optimization, and define the $p$-norm distance degree of $X$, which coincides with the ED degree of $X$ for $p=2$. Finally, we derive a formula for the $p$-norm distance degree of $X$ as a weighted sum of the polar classes of $X$ under suitable transversality conditions.

math.AG↗

arXiv2019

On the unique unexpected quartic in $\mathbb{P}^2$

The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in Algebraic Geometry. In general, it is still an open problem to understand when the points fail to impose independent conditions. Despite many partial results, a complete solution is not known, even if the fixed points are in general position. The answer in the case of general points in the projective plane is predicted by the famous Segre-Harbourne-Gimigliano-Hirschowitz conjecture. When we consider fixed points in special position, even more interesting situations may occur. Recently Di Gennaro, Ilardi and Vallès discovered a special configuration $Z$ of nine points with a remarkable property: a general triple point always fails to impose independent conditions on the ideal of $Z$ in degree four. The peculiar structure and properties of this kind of \textit{unexpected curves} were studied by Cook II, Harbourne, Migliore and Nagel. By using both explicit geometric constructions and more abstract algebraic arguments, we classify low degree unexpected curves. In particular, we prove that the aforementioned configuration $Z$ is the unique one giving rise to an unexpected quartic.

math.AG↗

arXiv2019

On the product of the singular values of a binary tensor

A real binary tensor consists of $2^d$ real entries arranged into hypercube format $2^{\times d}$. For $d=2$, a real binary tensor is a $2\times 2$ matrix with two singular values. Their product is the determinant. We generalize this formula for any $d\ge 2$. Given a partition $μ\vdash d$ and a $μ$-symmetric real binary tensor $t$, we study the distance function from $t$ to the variety $X_{μ,\mathbb{R}}$ of $μ$-symmetric real binary tensors of rank one. The study of the local minima of this function is related to the computation of the singular values of $t$. Denoting with $X_μ$ the complexification of $X_{μ,\mathbb{R}}$, the Euclidean Distance polynomial $\mathrm{EDpoly}_{X_μ^\vee,t}(ε^2)$ of the dual variety of $X_μ$ at $t$ has among its roots the singular values of $t$. On one hand, the lowest coefficient of $\mathrm{EDpoly}_{X_μ^\vee,t}(ε^2)$ is the square of the $μ$-discriminant of $t$ times a product of sum of squares polynomials. On the other hand, we describe the variety of $μ$-symmetric binary tensors that do not admit the maximum number of singular values, counted with multiplicity. Finally, we compute symbolically all the coefficients of $\mathrm{EDpoly}_{X_μ^\vee,t}(ε^2)$ for tensors of format $2\times 2\times 2$.

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