arXiv subjects

Lukas Kühne

Publications and source records attributed to Lukas Kühne.

At least 19 recordsLinked to original sources

arXiv2026

Combinatorial invariants of finite metric spaces and the Wasserstein arrangement

In 2010, Vershik proposed a new combinatorial invariant of metric spaces given by a class of polytopes that arise in the theory of optimal transport and are called ``Wasserstein polytopes'' or ``Kantorovich-Rubinstein polytopes'' in the literature. Answering a question posed by Vershik, we describe the stratification of the metric cone induced by the combinatorial type of these polytopes through a hyperplane arrangement. Moreover, we study its relationships with the stratification by combinatorial type of the injective hull (i.e., the tight span) and, in particular, with certain types of metrics arising in phylogenetic analysis. We also compute enumerative invariants in the case of metrics on up to six points.

math.CO↗

arXiv2026

Addition theorems for Ziegler pairs of hyperplane arrangements

Inspired by Terao's freeness conjecture, we examine Ziegler pairs, which are pairs of hyperplane arrangements that share the same underlying matroid but have different modules of logarithmic derivations. In this paper, we present a general construction that yields the first known families of Ziegler pairs in arbitrary dimension and size, starting from examples in the complex projective plane.

math.CO↗

arXiv2026

The $\mathbb{S}_n$-equivariant Chow polynomial of the braid matroid

We determine the generating function for the $\mathbb{S}_n$-equivariant Chow polynomials of the braid matroid $B_n$. The Chow polynomial of $B_n$ is the Poincaré polynomial of the wonderful compactification of the complement of the braid arrangement, with respect to the maximal building set. A key input to our result is the identification of this wonderful compactification with a moduli space of multiscale differentials, recently established by Devkota, Robotis, and Zahariuc. In this way, our work also contributes to the literature on topology of moduli spaces of multiscale differentials. To prove our main formula, we study the classes of these moduli spaces in the Grothendieck ring of varieties via the formalism of $\mathbb{S}_n$-spaces developed by Getzler and Pandharipande. We also give a new interpretation of the numerical Chow polynomial of $B_n$ as the Poincaré polynomial of a moduli space of genus-zero relative stable maps to $\mathbb{P}^1$.

math.AG↗

arXiv2026

Free line arrangements with low maximal multiplicity

Let $\A$ be a free arrangement of $d$ lines in the complex projective plane, with exponents $d_1\leq d_2$. Let $m$ be the maximal multiplicity of points in $\A$. In this note, we describe first the simple cases $d_1 \leq m$. Then we study the case $d_1=m+1$, and describe which line arrangements can occur by deleting or adding a line to $\A$. When $d \leq 14$, there are only two free arrangements with $d_1=m+2$, namely one with degree $13$ and the other with degree $14$. We study their geometries in order to deepen our understanding of the structure of free line arrangements in general.

math.AG↗

arXiv2025

Absolute incidence theorems and tilings

We give a precise definition of incidence theorems in plane projective geometry and introduce the notion of ``absolute incidence theorems,'' which hold over any ring. Fomin and Pylyavskyy describe how to obtain incidence theorems from tilings of an orientable surface; they call this result the ``master theorem''. Instances of the master theorem are always absolute incidence theorems. As most classically known incidence theorems are instances of the master theorem, they are absolute incidence theorems. We give an explicit example of an incidence theorem involving 13 points that is not an absolute incidence theorem, and therefore is not an instance of the master theorem.

math.CO↗

arXiv2025

Regular polygons, line operators, and elliptic modular surfaces as realization spaces of matroids

For an integer $n\geq 7$, we investigate the matroid realization space of a specific deformation of the regular $n$-gon along with its lines of symmetry. It turns out that this particular realization space is birational to the elliptic modular surface $Ξ_{1}(n)$ over the modular curve $X_{1}(n)$. In this way, we obtain a model of $Ξ_{1}(n)$ defined over the rational numbers. Furthermore, a natural geometric operator acts on these matroid realizations. On the elliptic modular surface, this operator corresponds to the multiplication by $-2$ on the elliptic curves. This provides a new geometric approach to computing multiplication by $-2$ on elliptic curves.

math.AG↗

arXiv2025

On the numerical Terao's conjecture and Ziegler pairs for line arrangements

In this paper we present a smallest possible counterexample to the Numerical Terao's Conjecture in the class of line arrangements in the complex projective plane. Our example consists of a pair of two arrangements with $13$ lines. Moreover, we use the newly discovered singular matroid realization spaces to construct new examples of pairs of line arrangements having the same underlying matroid but different free resolutions of the Milnor algebras. Such rare arrangements are called Ziegler pairs in the literature.

math.AG↗

arXiv2025

Dynamical systems on some elliptic modular surfaces via operators on line arrangements

This paper further studies the matroid realization space of a specific deformation of the regular $n$-gon with its lines of symmetry. Recently, we obtained that these particular realization spaces are birational to the elliptic modular surfaces $Ξ_{1}(n)$ over the modular curve $X_1(n)$. Here, we focus on the peculiar cases when $n=7,8$ in more detail. We obtain concrete quartic surfaces in $\mathbb{P}^3$ equipped with a dominant rational self-map stemming from an operator on line arrangements, which yields K3 surfaces with a dynamical system that is semi-conjugated to the plane.

math.AG↗

arXiv2025

Most $q$-matroids are not representable

A $q$-matroid is the analogue of a matroid which arises by replacing the finite ground set of a matroid with a finite-dimensional vector space over a finite field. These $q$-matroids are motivated by coding theory as the representable $q$-matroids are the ones that stem from rank-metric codes. In this note, we establish a $q$-analogue of Nelson's theorem in matroid theory by proving that asymptotically almost all $q$-matroids are not representable. This answers a question about representable $q$-matroids by Jurrius and Pellikaan strongly in the negative.

math.CO↗

arXiv2025

When alcoved polytopes add

Alcoved polytopes are characterized by the property that all facet normal directions are parallel to the roots $e_i-e_j$. Unlike other prominent families of polytopes, like generalized permutahedra, alcoved polytopes are not closed under Minkowski sums. We nonetheless show that the Minkowski sum of a collection of alcoved polytopes is alcoved if and only if each pairwise sum is alcoved. This implies that the type fan of alcoved polytopes is determined by its two-dimensional cones. Moreover, we provide a complete characterization of when the Minkowski sum of alcoved simplices is again alcoved via a graphical criterion on pairs of ordered set partitions. Our characterization reduces to checking conditions on restricted partitions of length at most six. In particular, we show how the Minkowski sum decompositions of the two most well-known families of alcoved polytopes, the associahedron and the cyclohedron, fit in our framework. Additionally, inspired by the physical construction of one-loop scattering amplitudes, we present a new infinite family of alcoved polytopes, called $\widehat{D}_n$ polytopes. We conclude by drawing a connection to matroidal blade arrangements and the Dressian.

math.CO↗

arXiv2025

On entropic and almost multilinear representability of matroids

This article studies two notions of generalized matroid representations motivated by algorithmic information theory and cryptographic secret sharing. The first (entropic representability) involves discrete random variables, while the second (almost-multilinear representability) deals with approximate subspace arrangements. In both cases, we prove that determining whether an input matroid has such a representation is undecidable. Consequently, the conditional independence implication problem is also undecidable, providing an independent answer to a question posed by Geiger and Pearl, recently resolved by Cheuk Ting Li. These problems are also closely related to characterizing achievable rates in network coding and constructing secret sharing schemes. For example, another corollary of our work is that deciding whether an access structure admits an ideal secret sharing scheme is undecidable. Our approach reduces undecidable problems from group theory to matroid representation problems. Specifically, we reduce the uniform word problem for finite groups to entropic representability and the word problem for sofic groups to almost-multilinear representability. A key part of this reduction involves modifying group presentations into forms where linear representations are generic in an appropriate sense when restricted to the generating set.

math.CO↗

arXiv2025

Arrangements and Likelihood

We develop novel tools for computing the likelihood correspondence of an arrangement of hypersurfaces in a projective space. This uses the module of logarithmic derivations. This object is well-studied in the linear case, when the hypersurfaces are hyperplanes. We here focus on nonlinear scenarios and their applications in statistics and physics.

math.AC↗

arXiv2024

Faces of Cosmological Polytopes

A cosmological polytope is a lattice polytope introduced by Arkani-Hamed, Benincasa, and Postnikov in their study of the wavefunction of the universe in a class of cosmological models. More concretely, they construct a cosmological polytope for any Feynman diagram, i.e. an undirected graph. In this paper, we initiate a combinatorial study of these polytopes. We give a complete description of their faces, identify minimal faces that are not simplices and compute the number of faces in specific instances. In particular, we give a recursive description of the $f$-vector of cosmological polytopes of trees.

math.CO↗

arXiv2024

Line arrangements with many triple points

In this paper, we construct an infinite series of line arrangements in characteristic two, each featuring only triple intersection points. This finding challenges the existing conjecture that suggests the existence of only a finite number of such arrangements, regardless of the characteristic. Leveraging the theory of matroids and employing computer algebra software, we rigorously examine the existence and non-existence across various characteristics of line arrangements with up to 19 lines maximizing the number of triple intersection points.

math.CO↗

arXiv2023

Matroids in OSCAR

OSCAR is an innovative new computer algebra system which combines and extends the power of its four cornerstone systems - GAP (group theory), Singular (algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic (number theory). Here, we present parts of the module handeling matroids in OSCAR, which will appear as a chapter of the upcoming OSCAR book. A matroid is a fundamental and actively studied object in combinatorics. Matroids generalize linear dependency in vector spaces as well as many aspects of graph theory. Moreover, matroids form a cornerstone of tropical geometry and a deep link between algebraic geometry and combinatorics. Our focus lies in particular on computing the realization space and the Chow ring of a matroid.

math.CO↗

arXiv2023

Projective dimension of weakly chordal graphic arrangements

A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. In this article we extend this result by proving that the module of logarithmic derivations of a graphic arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices.

math.CO↗

arXiv2022

Likelihood Degenerations

Computing all critical points of a monomial on a very affine variety is a fundamental task in algebraic statistics, particle physics and other fields. The number of critical points is known as the maximum likelihood (ML) degree. When the variety is smooth, it coincides with the Euler characteristic. We introduce degeneration techniques that are inspired by the soft limits in CEGM theory, and we answer several questions raised in the physics literature. These pertain to bounded regions in discriminantal arrangements and to moduli spaces of point configurations. We present theory and practise, connecting complex geometry, tropical combinatorics, and numerical nonlinear algebra.

math.AG↗

arXiv2022

On the geometry of flag Hilbert-Poincaré series for matroids

We extend the definition of coarse flag Hilbert--Poincaré series to matroids; these series arise in the context of local Igusa zeta functions associated to hyperplane arrangements. We study these series in the case of oriented matroids by applying geometric and combinatorial tools related to their topes. In this case, we prove that the numerators of these series are coefficient-wise bounded below by the Eulerian polynomial and equality holds if and only if all topes are simplicial. Moreover this yields a sufficient criterion for non-orientability of matroids of arbitrary rank.

math.CO↗