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Martin Vodička

Publications and source records attributed to Martin Vodička.

15 recordsLinked to original sources
arXiv2026

Self-inverse linear subspaces of matrices

We study linear subspaces of matrices whose inverse spaces are also linear. Based on the fact that any linear space containing the identity matrix and whose inverse space is linear must be a self-inverse space, we introduce such spaces as self-inverse spaces. In fact, as we will show, self-inverse spaces are finite-dimensional complex unitary Jordan algebras. We provide an algebraic classification of all self-inverse spaces of small type and a classification of small-dimensional self-inverse spaces up to isomorphism.

math.AG↗
arXiv2024

On the maximum likelihood degree for Gaussian graphical models

In this paper we revisit the likelihood geometry of Gaussian graphical models. We give a detailed proof that the ML-degree behaves monotonically on induced subgraphs. Furthermore, we complete a missing argument that the ML-degree of the $n$-th cycle is larger than one for any $n\geq 4$, therefore completing the characterization that the only Gaussian graphical models with rational maximum likelihood estimator are the ones corresponding to chordal (decomposable) graphs. Finally, we prove that the formula for the ML-degree of a cycle conjectured by Drton, Sturmfels and Sullivant provides a correct lower bound.

math.ST↗
arXiv2024

Phylogenetic degrees for Jukes-Cantor model

Jukes-Cantor model is one of the most meaningful statistical models from a biological perspective. We are interested in computing the algebraic degrees for phylogenetic varieties, which we call phylogenetic degrees, associated to the Jukes-Cantor model and any tree. As these varieties are toric, their geometry is hidden in the associated polytopes. For this reason, we provide two different combinatorial approaches to compute the volume for these polytopes.

math.AG↗
arXiv2023

Classification of normal phylogenetic varieties for tripods

We provide a complete classification of normal phylogenetic varieties coming from tripods, and more generally, from trivalent trees. Let $G$ be an abelian group. We prove that the group-based phylogenetic variety $X_{G,\mathcal{T}}$, for any trivalent tree $\mathcal{T}$, is projectively normal if and only if $G\in \{\mathbb{Z}_2, \mathbb{Z}_3, \mathbb{Z}_2\times\mathbb{Z}_2, \mathbb{Z}_4, \mathbb{Z}_5, \mathbb{Z}_7\}$.

math.AG↗
arXiv2023

Phylogenetic degrees for claw trees

Group-based models appear in algebraic statistics as mathematical models coming from evolutionary biology, respectively the study of mutations of organisms. Both theoretically and in terms of applications, we are interested in determining the algebraic degrees of the phylogenetic varieties coming from these models. These algebraic degrees are called phylogenetic degrees. In this paper, we compute the phylogenetic degree of the variety $X_{G, n}$ with $G\in\{\mathbb{Z}_2,\mathbb{Z}_2\times\mathbb{Z}_2, \mathbb{Z}_3\}$ and any $n$-claw tree. As these varieties are toric, computing their phylogenetic degree relies on computing the volume of their associated polytopes $P_{G,n}$. We apply combinatorial methods and we give concrete formulas for them.

math.AG↗
arXiv2021

Geometry of the Gaussian graphical model of the cycle

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 based on the intersection theory in the space of complete quadrics.

math.AG↗
arXiv2021

The leading coefficient of Lascoux polynomials

Lascoux polynomials have been recently introduced to prove polynomiality of the maximum-likelihood degree of linear concentration models. We find the leading coefficient of the Lascoux polynomials (type C) and their generalizations to the case of general matrices (type A) and skew symmetric matrices (type D). In particular, we determine the degrees of such polynomials. As an application, we find the degree of the polynomial $δ(m,n,n-s)$ of the algebraic degree of semidefinite programming, and when $s=1$ we find its leading coefficient for types C, A and D.

math.AG↗
arXiv2020

Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming

We establish connections between: the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices.

math.AG↗
arXiv2020

Gorenstein property for phylogenetic trivalent trees

We study the Gorenstein property for phylogenetic group-based models. We prove that for the groups $\mathbb Z_3$ and $\mathbb Z_2\times \mathbb Z_2$ and trivalent trees the associated polytopes are always Gorenstein extending the results of Buczyńska and Wiśniewski for the group $\mathbb Z_2$.

math.CO↗
arXiv2019

Gorenstein graphic matroids

The toric variety of a matroid is projectively normal, and therefore it is Cohen-Macaulay. We provide a complete graph-theoretic classification when the toric variety of a graphic matroid is Gorenstein.

math.CO↗
arXiv2019

Normality of the Kimura 3-parameter model

The Kimura 3-parameter model is one of the most fundamental phylogenetic models in algebraic statistics. We prove that all algebraic varieties associated to this model are projectively normal, confirming a conjecture of Michalek.

math.AG↗
arXiv2018

A uniform stability principle for dual lattices

We prove a highly uniform stability or "almost-near" theorem for dual lattices of lattices $L \subseteq \Bbb R^n$. More precisely, we show that, for a vector $x$ from the linear span of a lattice $L \subseteq \Bbb R^n$, subject to $λ_1(L) \ge λ> 0$, to be $\varepsilon$-close to some vector from the dual lattice $L'$ of $L$, it is enough that the inner products $u\,x$ are $δ$-close (with $δ< 1/3$) to some integers for all vectors $u \in L$ satisfying $\| u \| \le r$, where $r > 0$ depends on $n$, $λ$, $δ$ and $\varepsilon$, only. This generalizes an earlier analogous result proved for integral vector lattices by M. Mačaj and the second author. The proof is nonconstructive, using the ultraproduct construction and a slight portion of nonstandard analysis.

math.NT↗