arXiv subjects

Elias Tsigaridas

Publications and source records attributed to Elias Tsigaridas.

At least 19 recordsLinked to original sources

arXiv2026

Resultant multiplicity via projective degrees and applications to tensor eigenvalues

Given a system $\mathbf{f}=(f_1,\ldots,f_n)$ of $n$ homogeneous forms in $n$ variables of the same degree, Macaulay's resultant vanishes precisely when the polynomials have a common projective zero. Its order of vanishing measures the singularity of the resultant hypersurface at $\mathbf{f}$. In this paper, we study how this multiplicity reflects the geometry of the projective zero scheme defined by $\mathbf{f}$. We give an exact formula for the multiplicity, expressed in terms of the projective degrees of the rational map defined by $\mathbf{f}$. As a consequence, we obtain a geometric lower bound involving the degrees, dimensions, and multiplicities of the irreducible components of the projective zero scheme. This extends the multiplicity estimates of Roy and Ghidelli from zero-dimensional schemes to schemes of arbitrary dimension. Finally, we apply this geometric estimate to tensor eigenvalues. It translates directly into a lower bound for the algebraic multiplicity of a tensor eigenvalue in terms of the geometry of its eigenscheme. This settles a conjecture by Canino et al. and consequently settles earlier conjectures of Qi and of Hu and Ye concerning the relationship between algebraic, geometric, and span multiplicities of tensor eigenvalues.

math.AG↗

arXiv2026

Exact Algebraic Computation of Learning Coefficients for Two-Dimensional Singular Models

Classical information criteria such as the Bayesian Information Criterion (BIC) rely on regularity assumptions that break down for singular models, leading to incorrect model selection in settings such as deep learning. The Widely Applicable Bayesian Information Criterion (WBIC) relies on local learning coefficients $λ$, which in the analytic case coincides with local Real Log Canonical Thresholds (RLCT) of the Kullback-Leibler divergence of the model, to capture correct marginal likelihood asymptotics. Exact computation of the learning coefficients has been limited to special cases, and only sampling-based estimation methods are generally applicable. We present the first deterministic algorithm that computes local RLCTs exactly for any two-dimensional model whose Kullback-Leibler distance is contact equivalent to a polynomial, derive a bound on its complexity, and demonstrate its effectiveness for a broad class of models, with applications including polynomial neural networks. Beyond providing ground truth to calibrate sampling-based estimators, exact computation reveals algebraic structure in learning coefficients that sampling cannot and out-speeds it in the shallow regime.

cs.LG↗

arXiv2026

Small Resultant Systems via Linear Combinations

For a system of $s$ homogeneous polynomials of degree $d$ in $n$ variables, say ${\bf{f}} = 0$, we consider the problem of constructing resultant systems. A resultant system is a finite set of polynomials in the coefficients of the input polynomials, the vanishing of which characterizes the systems $\bf{f}$ with a common non-zero solution. The classical approaches for constructing resultant systems rely either on maximal minors of large coefficient matrices or on the coefficients of a resultant of generic linear combinations of the input polynomials. Typically, they produce resultant systems containing a very large number of polynomials. We develop new constructions based on taking resultants of linear combinations of the input polynomials; this results in resultant systems of small cardinality. Our main results are: 1) We prove that a resultant system with ${d+n-1 \choose n-1} s-n^2+1$ polynomials exists; each polynomial is the resultant of $n$ linear combinations of the input polynomials. This improves the previously known upper bounds, even for systems of bivariate homogeneous polynomials. 2) Under the assumption that the input polynomials are non-zero, we construct explicit resultant systems with cardinality $\mathrm{poly}(s,d)$, when $n$ is fixed.

math.AC↗

arXiv2026

On the equivalence of semidefinite programming and zero-sum semidefinite games

By results of Dantzig (1951) and Adler (2013), computing the optimal solutions of a linear program is equivalent to finding optimal strategies in zero-sum bimatrix games. Dantzig's original result was incomplete, in the sense that the reduction of a linear program to a zero-sum game did not work for all possible linear programs. We show that, under a natural constraint qualification requiring either the existence of strongly optimal primal-dual solutions or of a strictly unbounded direction, computing the solution of a semidefinite program is equivalent to finding optimal strategies in an associated zero-sum semidefinite game. Our work builds upon Ickstadt, Theobald, and Tsigaridas (2024), where, similar to Dantzig's work, the proposed reduction cannot handle a certain subclass of semidefinite programs. Our main proof ingredients for the equivalence result include: (i) a semidefinite generalization of von Stengel's (2023) extension of Dantzig's construction; (ii) techniques for handling more general duality phenomena in the semidefinite setting; and (iii) an explicit bound for the (coordinates) of the solutions of a semidefinite program. As a by-product, the game value provides a certificate: it is zero if and only if strongly optimal solutions exist, and otherwise optimal strategies yield an infeasibility certificate for the primal or dual program.

math.OC↗

arXiv2025

Positive Univariate Polynomials: SOS certificates, algorithms, bit complexity, and T-systems

We consider certificates of positivity for univariate polynomials with rational coefficients that are positive over (an interval of)~$\mathbb{R}$. Such certificates take the form of weighted sums of squares (SOS) of polynomials with rational coefficients. We build on the algorithm of Chevillard, Harrison, Jolde{ş}, and Lauter~\cite{chml-usos-alg-11}, and we introduce a variant that we refer to as \usos. Given a polynomial of degree~$d$ with maximum coefficient bitsize~$τ$, we show that \usos computes a rational weighted SOS representation in $\widetilde{\mathcal{O}}_B(d^3 + d^2 τ)$ bit operations; the resulting certificate of posivitity involves rationals of bitsize $\widetilde{\mathcal{O}}(d^2 τ)$. This improves the best-known complexity bounds by a factor of~$d$ and completes previous analyses. We also extend these results to certificates of positivity over arbitrary rational intervals, via a simple transformation. In this case as well, our techniques yield a factor-$d$ improvement in the complexity bounds. Along the same line, for univariate polynomials with rational coefficients, we introduce a new class of certificates, which we call \emph{perturbed SOS certificates}. They consist of a sum of two rational squares that approximates the input polynomial closely enough so that nonnegativity of the approximation implies the nonnegativity of the original polynomial. This computation has the same bit complexity and yields certificates of the same bitsize as in the weighted SOS case. We further investigate structural properties of these SOS decompositions. Relying on the classical result that any nonnegative univariate real polynomial is the sum of two squares of real polynomials, we show that the summands form an interlacing pair. Consequently, their real roots correspond to the Karlin points of the original polynomial on~$\mathbb{R}$, establishing a new connection with the T-systems studied by Karlin~\cite{Karlin-repr-pos-63}. This connection enables us to compute such decompositions explicitly. Previously, only existential results were known for T-systems. We obtain analogous results for positivity over $(0, \infty)$, and hence over arbitrary real intervals. Finally, we present our open-source Maple implementation of the \usos algorithm, together with experiments on various data sets demonstrating the efficiency of our approach.

cs.CC↗

arXiv2025

Semidefinite network games: multiplayer minimax and complementarity problems

Network games provide a powerful framework for modeling agent interactions in networked systems, where players are represented by nodes in a graph and their payoffs depend on the actions taken by their neighbors. Extending the framework of network games, we introduce and study semidefinite network games. In this model, each player selects a positive semidefinite matrix with trace equal to one, known as a density matrix, to engage in a two-player game with every neighboring node. The player's payoff is the cumulative payoff acquired from these edge games. Initially, we focus on the zero-sum setting, where the sum of all players' payoffs is equal to zero. We establish that, in this class of games, Nash equilibria can be characterized as the projection of a spectrahedron. Furthermore, we show that determining whether a semidefinite network game is a zero-sum game is equivalent to deciding if the value of a semidefinite program is zero. Beyond the zero-sum case, we characterize Nash equilibria as the solutions of a semidefinite linear complementarity problem.

math.OC↗

arXiv2025

Certificates for nonnegativity of multivariate integer polynomials under perturbations

We develop a general and unconditional framework for certifying the global nonnegativity of multivariate integer polynomials; based on rewriting them as sum of squares modulo their gradient ideals. We remove the two structural assumptions typically required by other approaches, namely that the polynomial attains its infimum and zero-dimensionality of the gradient ideal. Our approach combines a denominator-free stereographic transformation with a refined variant of the Hanzon--Jibetean perturbation scheme. The stereographic transformation preserves nonnegativity while making the polynomial coercive, with explicit bounds on the radius of positivity and on the nonzero critical values. Subsequently, we apply carefully constructed explicit perturbations that enforce zero-dimensionality of the gradient ideal without altering nonnegativity, allowing us to invoke recent algorithms to derive algebraic certificates or rational witness points. We present three algorithms implementing our framework and analyze their bit complexity in detail, which is single exponential with respect to the number of variables. A second contribution is a new explicit SOS perturbation scheme, which allows us to perturb any nonnegative polynomial in such a way that it can be written as a sum of squares (SOS). In contrast to Lasserre's classical SOS approximation, which guaranties density but currently does not provide an effective control over the perturbation size, we only derive concrete perturbation bounds ensuring that a nonnegative polynomial enters the SOS cone.

cs.SC↗

arXiv2025

Optimal Preconditioning is a Geodesically Convex Optimization Problem

We introduce a unified framework for computing approximately-optimal preconditioners for solving linear and non-linear systems of equations. We demonstrate that the condition number minimization problem, under structured transformations such as diagonal and block-diagonal preconditioners, is geodesically convex with respect to unitarily invariant norms, including the Frobenius and Bombieri--Weyl norms. This allows us to introduce efficient first-order algorithms with precise convergence guarantees. For linear systems, we analyze the action of symmetric Lie subgroups $G \subseteq \GL_m(\CC) \times \GL_n(\CC)$ on the input matrix and prove that the logarithm of the condition number is a smooth geodesically convex function on the associated Riemannian quotient manifold. We obtain explicit gradient formulas, show Lipschitz continuity, and prove convergence rates for computing the optimal Frobenius condition number: $\widetilde{O}(1/\eps^2)$ iterations for general two-sided preconditioners and $\widetilde{O}(κ_F^2 \log(1/\eps))$ for strongly convex cases such as left preconditioning. We extend our framework to consider preconditioning of polynomial systems $\f(x) = 0$, where $\f$ is a system of multivariate polynomials. We analyze the local condition number $μ(\f, ξ)$, at a root $ξ$ and prove that it also admits a geodesically convex formulation under appropriate group actions. We deduce explicit formulas for the Riemannian gradients and present convergence bounds for the corresponding optimization algorithms. To the best of our knowledge, this is the first preconditioning algorithm with theoretical guarantees for polynomial systems.

math.OC↗

arXiv2025

Solving bihomogeneous polynomial systems with a zero-dimensional projection

We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the corresponding quotient ring, we introduce linear maps that greatly extend the classical multiplication maps for zero-dimensional systems, but are not those associated to the elimination ideal; we also call them multiplication maps. We construct them using linear algebra on the restriction of the ideal to a carefully chosen bidegree or, if available, from an arbitrary Gröbner bases. The multiplication maps allow us to compute the elimination ideal of the projection, by generalizing FGLM algorithm to bihomogenous, non-zero dimensional, varieties. We also study their properties, like their minimal polynomials and the multiplicities of their eigenvalues, and show that we can use the eigenvalues to compute numerical approximations of the zero-dimensional projection. Finally, we establish a single exponential complexity bound for computing multiplication maps and Gröbner bases, that we express in terms of the bidegrees of the generators of the corresponding bihomogeneous ideal.

math.AC↗

arXiv2025

Nash equilibria in semidefinite games and Lemke-Howson paths

We consider an algorithmic framework for two-player non-zero-sum semidefinite games, where each player's strategy is a positive semidefinite matrix with trace one. We formulate the computation of Nash equilibria in such games as semidefinite complementarity problems and develop symbolic-numeric techniques to trace generalized Lemke-Howson paths. These paths generalize the piecewise affine-linear trajectories of the classical Lemke-Howson algorithm for bimatrix games, replacing them with nonlinear curve branches governed by eigenvalue complementarity conditions. A key feature of our framework is the introduction of event points, which correspond to curve singularities. We analyze the local behavior near these points using Puiseux series expansions. We prove the smoothness of the curve branches under suitable non-degeneracy conditions and establish connections between our approach and both the classical combinatorial and homotopy-theoretic interpretations of the Lemke-Howson algorithm.

math.OC↗

arXiv2025

Beyond Worst-Case Analysis for Symbolic Computation: Root Isolation Algorithms

We introduce beyond-worst-case analysis into symbolic computation. This is an extensive field which almost entirely relies on worst-case bit complexity, and we start from a basic problem in the field: isolating the real roots of univariate polynomials. This is a fundamental problem in symbolic computation and it is arguably one of the most basic problems in computational mathematics. The problem has a long history decorated with numerous ingenious algorithms and furnishes an active area of research. However, most available results in literature either focus on worst-case analysis in the bit complexity model or simply provide experimental benchmarking without any theoretical justifications of the observed results. We aim to address the discrepancy between practical performance of root isolation algorithms and prescriptions of worst-case complexity theory: We develop a smoothed analysis framework for polynomials with integer coefficients to bridge this gap. We demonstrate (quasi-)linear (expected and smoothed) complexity bounds for Descartes algorithm, that is one most well know symbolic algorithms for isolating the real roots of univariate polynomials with integer coefficients. Our results explain the surprising efficiency of Descartes solver in comparison to sophisticated algorithms that have superior worst-case complexity. We also analyse the Sturm solver, ANewDsc a symbolic-numeric algorithm that combines Descartes with Newton operator, and a symbolic algorithm for sparse polynomials.

cs.SC↗

arXiv2025

Bigraded Castelnuovo-Mumford regularity and Gröbner bases

We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal $I$ in the coordinate ring of the product of two projective spaces and the bidegrees of a Gröbner basis of $I$ with respect to the degree reverse lexicographical monomial order in generic coordinates. For the single-graded case, Bayer and Stillman unraveled all aspects of this relationship forty years ago and these results led to complexity estimates for computations with Gröbner bases. We build on this work to introduce a bounding region of the bidegrees of minimal generators of bihomogeneous Gröbner bases for $I$. We also use this region to certify the presence of some minimal generators close to its boundary. Finally, we show that, up to a certain shift, this region is related to the bigraded Castelnuovo-Mumford regularity of $I$.

math.AC↗

arXiv2024

Stratification of Projection Maps From Toric Varieties

We prove a combinatorial version of Thom's Isotopy Lemma for projection maps applied to any complex or real toric variety. Our results are constructive and give rise to a method for associating the Whitney strata of the projection to the faces of the polytope of the corresponding toric variety. For all examples we produced, our resulting algorithm outperforms known general purpose methods in Helmer and Nanda (FoCM, 2022), and Dinh and Jelonek (DCG, 2021) for computing map-stratifications.

math.AG↗

arXiv2024

Bounds on the infimum of polynomials over a generic semi-algebraic set using asymptotic critical values

We present precise bit and degree estimates for the optimal value of the polynomial optimization problem $f^*:=\text{inf}_{x\in \mathscr{X}}~f(x)$, where $\mathscr{X}$ is a semi-algebraic set satisfying some non-degeneracy conditions. Our bounds depend on the degree, the bitsize of $f$, and the polynomials defining $\mathscr{X}$, and are single exponential with respect to the number of variables. They generalize the single exponential bounds from Jeronimo, Perrucci, and Tsigaridas (SIAM Journal on Optimization, 23(1):241--255, 2013) for the minimum of a polynomial function on a compact connected component of a basic closed semi-algebraic set. The tools that we use allow us to obtain specialized bounds and dedicated algorithms for two large families of polynomial optimization problems in which the optimum value might not be attained. The first family forms a dense set of real polynomial functions with a fixed collection of Newton polytopes; we provide the best approximation yet for the bifurcation set, which contains the optimal value, and we deduce an effective method for computations. As for the second family, we consider any unconstrained polynomial optimization problem; we present more precise bounds, together with a better bit complexity estimate of an algorithm to compute the optimal value.

math.OC↗

arXiv2024

Semidefinite games

We introduce and study the class of semidefinite games, which generalizes bimatrix games and finite $N$-person games, by replacing the simplex of the mixed strategies for each player by a slice of the positive semidefinite cone in the space of real symmetric matrices. For semidefinite two-player zero-sum games, we show that the optimal strategies can be computed by semidefinite programming. Furthermore, we show that two-player semidefinite zero-sum games are almost equivalent to semidefinite programming, generalizing Dantzig's result on the almost equivalence of bimatrix games and linear programming. For general two-player semidefinite games, we prove a spectrahedral characterization of the Nash equilibria. Moreover, we give constructions of semidefinite games with many Nash equilibria. In particular, we give a construction of semidefinite games whose number of connected components of Nash equilibria exceeds the long standing best known construction for many Nash equilibria in bimatrix games, which was presented by von Stengel in 1999.

math.OC↗

arXiv2024

On the complexity of Chow and Hurwitz forms

We consider the bit complexity of computing Chow forms and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model, and our result represents the first bit complexity bound. We also extend our algorithm to Hurwitz forms in projective space, and explore connections between multiprojective Hurwitz forms and matroid theory. The motivation for our work comes from incidence geometry where intriguing computational algebra problems remain open.

cs.CC↗

arXiv2024

Randomized Control in Performance Analysis and Empirical Asset Pricing

The present article explores the application of randomized control techniques in empirical asset pricing and performance evaluation. It introduces geometric random walks, a class of Markov chain Monte Carlo methods, to construct flexible control groups in the form of random portfolios adhering to investor constraints. The sampling-based methods enable an exploration of the relationship between academically studied factor premia and performance in a practical setting. In an empirical application, the study assesses the potential to capture premias associated with size, value, quality, and momentum within a strongly constrained setup, exemplified by the investor guidelines of the MSCI Diversified Multifactor index. Additionally, the article highlights issues with the more traditional use case of random portfolios for drawing inferences in performance evaluation, showcasing challenges related to the intricacies of high-dimensional geometry.

q-fin.PM↗

arXiv2023

Computing the non-properness set of real polynomial maps in the plane

We introduce novel mathematical and computational tools to develop a complete algorithm for computing the set of non-properness of polynomials maps in the plane. In particular, this set, which we call \emph{the Jelonek set}, is a subset of $\mathbb{K}^2$ where a dominant polynomial map $f:\mathbb{K}^2\to\mathbb{K}^2$ is not proper; $\mathbb{K}$ could be either $\mathbb{C}$ or $\mathbb{R}$. Unlike all the previously known approaches we make no assumptions on $f$ whenever $\mathbb{K} = \mathbb{R}$; this is the first algorithm with this property. The algorithm takes into account the Newton polytopes of the polynomials. As a byproduct we provide a finer representation of the set of non-properness as a union of semi-algebraic curves, that correspond to edges of the Newton polytopes, which is of independent interest. Finally, we present a precise Boolean complexity analysis of the algorithm and a prototype implementation in Maple.

math.AG↗