arXiv subjects

Tobias Boege

Publications and source records attributed to Tobias Boege.

At least 19 recordsLinked to original sources

arXiv2026

Trinomial containment in polynomial ideals is undecidable

We prove that deciding whether an ideal in a polynomial ring contains a trinomial is impossible on a Turing machine. More precisely, from an integer polynomial $P$ we compute generators of an ideal $I_P$ in a polynomial ring over $\mathbb{Q}$ such that $I_P$ contains a trinomial if and only if $P$ has an integral zero. By the MRDP theorem this problem is undecidable. A universal halting polynomial gives a computable family of ideals in one fixed polynomial ring, with uniform bounds on colength, generator count, and generator degree, for which the containment of a trinomial encodes the halting problem.

math.AC↗

arXiv2026

Recognition of algebraic matroids is undecidable

We prove that the recognition problem for algebraic matroids is undecidable. Explicitly, this means that there is no algorithm that takes as input a finite set $S$ and a function $r\colon\mathcal{P}(S) \to \mathbb{Z}_{\ge 0}$ (where $\mathcal{P}(S)$ is the power set) and decides whether there exists a pair of fields $F \subset K$, and a function $f\colon S \to K$, such that for all $A \subseteq S$: $\mathrm{trdeg}_F f(A) = r(A)$. This problem is known to be decidable if the characteristic of the fields involved is constrained to be zero. We prove that it is undecidable if the characteristic is either left unspecified (in which case a realization over any characteristic is accepted) or fixed to be a prime $p$. The proof relies on Hrushovski--Zilber's Group Configuration Theorem and on the work of Evans and Hrushovski on "Projective Planes in Algebraically Closed Fields". We relate two different such projective planes, and eventually construct a reduction from the solvability of Diophantine equations over $\mathbb{F}_p(x)$ ($p$ prime) to algebraicity of matroids. Solvability of Diophantine equations over $\mathbb{F}_p(x)$ was proved to be undecidable by Pheidas for all $p > 2$, and later by Videla for $p=2$. A central part of our proof is a variant of the so-called Field Configuration Theorem.

math.CO↗

arXiv2026

Sharp Inequalities for Products of Principal Minors of Positive Definite Matrices

We study sharp inequalities for ratios of products of principal minors of real positive definite matrices. Our main result gives a closed-form solution to a family of nonconvex optimization problems over the positive definite cone. As a special case, we prove that the infimum of the Ingleton ratio over $4\times 4$ positive definite matrices is $16/27$, confirming a conjecture of Hall and Johnson. We also show that the cone of absolutely bounded ratios of products of principal minors is not polyhedral for $n\ge 4$, and that it is not semialgebraic over $\mathbb{Q}$.

math.MG↗

arXiv2026

On the Intersection and Composition properties of conditional independence

Compositional graphoids are fundamental discrete structures which appear in probabilistic reasoning, particularly in the area of graphical models. They are semigraphoids which satisfy the Intersection and Composition properties. These important properties, however, are not enjoyed by general probability distributions. This paper surveys what is known about them, providing systematic constructions of examples and counterexamples as well as necessary and sufficient conditions. Novel sufficient conditions for both properties are derived in the context of discrete random variables via information-theoretic tools.

cs.IT↗

arXiv2026

Polyhedral Aspects of Maxoids

The conditional independence (CI) relation of a distribution in a max-linear Bayesian network depends on its weight matrix through the $C^\ast$-separation criterion. These CI~models, which we call maxoids, are compositional graphoids which are in general not representable by Gaussian random variables. We prove that every maxoid can be obtained from a transitively closed weighted DAG and show that the stratification of generic weight matrices by their maxoids yields a polyhedral~fan. We also use this connection to polyhedral geometry to develop an algorithm for solving the conditional independence implication problem for maxoids.

math.CO↗

arXiv2026

Algebraic Statistics in OSCAR

We introduce the AlgebraicStatistics section of the OSCAR computer algebra system. We give an overview of its extensible design and highlight its features including serialization of data types for sharing results and creating databases, and state-of-the-art implicitization algorithms.

stat.CO↗

arXiv2025

Colored Gaussian directed acyclic graphical models

We study submodels of Gaussian DAG models defined by partial homogeneity constraints imposed on the model error variances and structural coefficients. We represent these models with colored DAGs and investigate their properties for use in statistical and causal inference. Local and global Markov properties are provided and shown to characterize the colored DAG model. Additional properties relevant to causal discovery are studied, including the existence and non-existence of faithful distributions and structural identifiability. Extending prior work of Peters and Bühlmann and Wu and Drton, we prove structural identifiability under the assumption of homogeneous structural coefficients, as well as for a family of models with partially homogeneous structural coefficients. The latter models, termed BPEC-DAGs, capture additional causal insights by clustering the direct causes of each node into communities according to their effect on their common target. An analogue of the GES algorithm for learning BPEC-DAGs is given and evaluated on real and synthetic data. Regarding model geometry, we provide a proof of a conjecture of Sullivant which generalizes to colored DAG models, colored undirected graphical models and directed ancestral graph models. The proof yields a tool for identification of Markov properties for any rationally parameterized model with globally, rationally identifiable parameters.

math.ST↗

arXiv2025

Structural Identifiability of Graphical Continuous Lyapunov Models

We prove two characterizations of model equivalence of acyclic graphical continuous Lyapunov models (GCLMs) with uncorrelated noise. The first result shows that two graphs are model equivalent if and only if they have the same skeleton and equivalent induced 4-node subgraphs. We also give a transformational characterization via structured edge reversals. The two theorems are Lyapunov analogues of celebrated results for Bayesian networks by Verma and Pearl, and Chickering, respectively. Our results have broad consequences for the theory of causal inference of GCLMs. First, we find that model equivalence classes of acyclic GCLMs refine the corresponding classes of Bayesian networks. Furthermore, we obtain polynomial-time algorithms to test model equivalence and structural identifiability of given directed acyclic graphs.

math.ST↗

arXiv2025

The entropy profiles of a definable set over finite fields

A definable set $X$ in the first-order language of rings defines a family of random vectors: for each finite field $\mathbb{F}_q$, let the distribution be supported and uniform on the $\mathbb{F}_q$-rational points of $X$. We employ results from the model theory of finite fields to show that their entropy profiles settle into one of finitely many stable asymptotic behaviors as $q$ grows. The attainable asymptotic entropy profiles and their dominant terms as functions of $q$ are computable. This generalizes a construction of Matúš which gives an information-theoretic interpretation to algebraic matroids.

cs.IT↗

arXiv2025

Sign patterns of principal minors of real symmetric matrices

We analyze a combinatorial rule satisfied by the signs of principal minors of a real symmetric matrix. The sign patterns satisfying this rule are equivalent to uniform oriented Lagrangian matroids. We first discuss their structure and symmetries and then study their asymptotics, proving that almost all of them are not representable by real symmetric matrices. We offer several conjectures and experimental results concerning representable sign patterns and the topology of their representation spaces.

math.CO↗

arXiv2024

The geometry of Gaussian double Markovian distributions

Gaussian double Markovian models consist of covariance matrices constrained by a pair of graphs specifying zeros simultaneously in the covariance matrix and its inverse. We study the semi-algebraic geometry of these models, in particular their dimension, smoothness and connectedness as well as algebraic and combinatorial properties.

math.ST↗

arXiv2024

Self-adhesivity in lattices of abstract conditional independence models

We introduce an algebraic concept of the frame for abstract conditional independence (CI) models, together with basic operations with respect to which such a frame should be closed: copying and marginalization. Three standard examples of such frames are (discrete) probabilistic CI structures, semi-graphoids and structural semi-graphoids. We concentrate on those frames which are closed under the operation of set-theoretical intersection because, for these, the respective families of CI models are lattices. This allows one to apply the results from lattice theory and formal concept analysis to describe such families in terms of implications among CI statements. The central concept of this paper is that of self-adhesivity defined in algebraic terms, which is a combinatorial reflection of the self-adhesivity concept studied earlier in context of polymatroids and information theory. The generalization also leads to a self-adhesivity operator defined on the hyper-level of CI frames. We answer some of the questions related to this approach and raise other open questions. The core of the paper is in computations. The combinatorial approach to computation might overcome some memory and space limitation of software packages based on polyhedral geometry, in particular, if SAT solvers are utilized. We characterize some basic CI families over 4 variables in terms of canonical implications among CI statements. We apply our method in information-theoretical context to the task of entropic region demarcation over 5 variables.

math.CO↗

arXiv2024

Real birational implicitization for statistical models

We derive an implicit description of the image of a semialgebraic set under a birational map, provided that the denominators of the map are positive on the set. For statistical models which are globally rationally identifiable, this yields model-defining constraints which facilitate model membership testing, representation learning, and model equivalence tests. Many examples illustrate the applicability of our results. The implicit equations recover well-known Markov properties of classical graphical models, as well as other well-studied equations such as the Verma constraint. They also provide Markov properties for generalizations of these frameworks, such as colored or interventional graphical models, staged trees, and the recently introduced Lyapunov models. Under a further mild assumption, we show that our implicit equations generate the vanishing ideal of the model up to a saturation, generalizing previous results of Geiger, Meek and Sturmfels, Duarte and Görgen, Sullivant, and others.

math.ST↗

arXiv2024

Conditional Independence in Stationary Diffusions

Stationary distributions of multivariate diffusion processes have recently been proposed as probabilistic models of causal systems in statistics and machine learning. Motivated by these developments, we study stationary multivariate diffusion processes with a sparsely structured drift. Our main result gives a characterization of the conditional independence relations that hold in a stationary distribution. The result draws on a graphical representation of the drift structure and pertains to conditional independence relations that hold generally as a consequence of the drift's sparsity pattern.

math.ST↗

arXiv2024

No eleventh conditional Ingleton inequality

A rational probability distribution on four binary random variables $X, Y, Z, U$ is constructed which satisfies the conditional independence relations $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Y]$, $[X \mathrel{\text{$\perp\mkern-10mu\perp$}} Z \mid U]$, $[Y \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid Z]$ and $[Z \mathrel{\text{$\perp\mkern-10mu\perp$}} U \mid XY]$ but whose entropy vector violates the Ingleton inequality. This settles a recent question of Studený (IEEE Trans. Inf. Theory vol. 67, no. 11) and shows that there are, up to symmetry, precisely ten inclusion-minimal sets of conditional independence assumptions on four discrete random variables which make the Ingleton inequality hold. The last case in the classification of which of these inequalities are essentially conditional is also settled.

cs.IT↗

arXiv2023

Selfadhesivity in Gaussian conditional independence structures

Selfadhesivity is a property of entropic polymatroids which guarantees that the polymatroid can be glued to an identical copy of itself along arbitrary restrictions such that the two pieces are independent given the common restriction. We show that positive definite matrices satisfy this condition as well and examine consequences for Gaussian conditional independence structures. New axioms of Gaussian CI are obtained by applying selfadhesivity to the previously known axioms of structural semigraphoids and orientable gaussoids.

cs.IT↗

arXiv2022

Research-Data Management Planning in the German Mathematical Community

In this paper we discuss the notion of research data for the field of mathematics and report on the status quo of research-data management and planning. A number of decentralized approaches are presented and compared to needs and challenges faced in three use cases from different mathematical subdisciplines. We highlight the importance of tailoring research-data management plans to mathematicians' research processes and discuss their usage all along the data life cycle.

math.HO↗

arXiv2022

Algebra in probabilistic reasoning

This short expository paper outlines applications of computer algebra to the implication problem of conditional independence for Gaussian random variables. We touch on certificates for validity and invalidity of inference rules from the perspective of reproducibility of research data, computational complexity of the inference problem and draw a parallel to automated theorem proving in synthetic geometry.

math.ST↗