40 citations · 460 across the 64 of their papers we have counts for
77 papers
A counterexample to Purdy's inequality for hyperplane arrangements in projective three-space
Mateusz Michałek, Piotr Pokora
We record an explicit counterexample to a refined form of Purdy's inequality for essential hyperplane arrangements in projective three-space. Let be an arrangement of…
Sullivant-Talaska ideal of the cyclic Gaussian Graphical Model
Austin Conner, Kangjin Han, Mateusz Michałek
In this paper, we settle a conjecture due to Sturmfels and Uhler concerning generation of the prime ideal of the variety associated to the Gaussian graphical model of any cycle gra…
Khovanskii bases for semimixed systems of polynomial equations -- a case of approximating stationary nonlinear Newtonian dynamics
Viktoriia Borovik, Paul Breiding, Javier del Pino +2
We provide an approach to counting roots of polynomial systems, where each polynomial is a general linear combination of prescribed, fixed polynomials. Our tools rely on the theory…
Polynomial systems admitting a simultaneous solution
Austin Conner, Mateusz Michalek, Michael Schindler +1
We provide a complete description of the ideal that serves as the resultant ideal for n univariate polynomials of degree d. We in particular describe a set of generators of this re…
Equivariant Euler characteristics on permutohedral varieties
Vincenzo Galgano, Hanieh Keneshlou, Mateusz Michalek
By the work of J.Huh, one can interpret binomial coefficients as a solution to an intersection problem on a permutohedral variety . Applying Hirzebruch-Riemann-Roch, this inte…
Enumerative geometry meets statistics, combinatorics and topology
Mateusz Michałek
We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on di…
The Algebraic Degree of Coupled Oscillators
Paul Breiding, Mateusz Michałek, Leonid Monin +1
Approximating periodic solutions to the coupled Duffing equations amounts to solving a system of polynomial equations. The number of complex solutions measures the algebraic comple…
Identifiability for mixtures of centered Gaussians and sums of powers of quadratics
Alexander Taveira Blomenhofer, Alex Casarotti, Alessandro Oneto +1
We consider the inverse problem for the polynomial map which sends an -tuple of quadratic forms in variables to the sum of their -th powers. This map captures the moment…
Geometry of the Gaussian graphical model of the cycle
Rodica Andreea Dinu, Mateusz Michałek, Martin Vodička
We prove a conjecture due to Sturmfels and Uhler concerning the degree of the projective variety associated to the Gaussian graphical model of the cycle. We involve new methods bas…
Applications of intersection theory: from maximum likelihood to chromatic polynomials
Rodica Andreea Dinu, Mateusz Michałek, Tim Seynnaeve
Recently, we have witnessed tremendous applications of algebraic intersection theory to branches of mathematics, that previously seemed very distant. In this article we review some…
Characteristic numbers and chromatic polynomial of a tensor
Austin Conner, Mateusz Michałek
We introduce the characteristic numbers and the chromatic polynomial of a tensor. Our approach generalizes and unifies the chromatic polynomial of a graph and of a matroid, charact…
Equations for GL invariant families of polynomials
Paul Breiding, Christian Ikenmeyer, Mateusz Michałek +1
We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer…
Algebraic compressed sensing
Paul Breiding, Fulvio Gesmundo, Mateusz Michałek +1
We introduce the broad subclass of algebraic compressed sensing problems, where structured signals are modeled either explicitly or implicitly via polynomials. This includes, for i…
Ideals of Spaces of Degenerate Matrices
Julian Vill, Mateusz Michałek, Alexander Taveira Blomenhofer
The variety consists of all tuples of matrices such that every linear combination of is singula…
Bounds on complexity of matrix multiplication away from CW tensors
Roser Homs, Joachim Jelisiejew, Mateusz Michałek +1
We present three families of minimal border rank tensors: they come from highest weight vectors, smoothable algebras, or monomial algebras. We analyse them using Strassen's laser m…
A note on seminormality of cut polytopes
Michał Lasoń, Mateusz Michałek
We prove that seminormality of cut polytopes is equivalent to normality. This settles two conjectures regarding seminormality of cut polytopes.
Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming
Laurent Manivel, Mateusz Michałek, Leonid Monin +2
We establish connections between: the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert ca…
Border rank non-additivity for higher order tensors
Matthias Christandl, Fulvio Gesmundo, Mateusz Michałek +1
Whereas matrix rank is additive under direct sum, in 1981 Schönhage showed that one of its generalizations to the tensor setting, tensor border rank, can be strictly subadditive fo…
Gorenstein matroids
Michał Lasoń, Mateusz Michałek
Matroid base polytope and independence polytope are Cohen-Macaulay. There are plenty of examples when they are also Gorenstein. We provide a full classification of matroids whose i…
Maximum likelihood degree, complete quadrics and -action
Mateusz Michałek, Leonid Monin, Jarosław Wiśniewski
We study the maximum likelihood (ML) degree of linear concentration models in algebraic statistics. We relate it to an intersection problem on the variety of complete quadrics. Thi…
Towards finding hay in a haystack: explicit tensors of border rank greater than in
J. M. Landsberg, Mateusz Michałek
We write down an explicit sequence of tensors in , for all sufficiently large, having border rank at least , overcoming a longstanding barrier…
Vanishing Hessian, wild forms and their border VSP
Hang Huang, Mateusz Michałek, Emanuele Ventura
Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between…
Many faces of symmetric edge polytopes
Alessio D'Alì, Emanuele Delucchi, Mateusz Michałek
Symmetric edge polytopes are a class of lattice polytopes constructed from finite simple graphs. In the present paper we highlight their connections to the Kuramoto synchronization…
Secant varieties of toric varieties arising from simplicial complexes
M. Azeem Khadam, Mateusz Michałek, Piotr Zwiernik
Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affin…