Tootfinder

Combinatorics of slices of cubes
Marie-Charlotte Brandenburg, Chiara Meroni
https://arxiv.org/abs/2510.09265 https://arxiv.org/pdf/2510.09265

Combinatorics of slices of cubes
We present a complete computational classification of the combinatorial types of hyperplane sections, or slices, of the regular cube up to dimension six. For each dimension, we determine the exact number of distinct combinatorial types. When restricted to slices through the origin, our computations extend to dimension seven. The classification combines combinatorial, algebraic, and numerical techniques, with all results certified. Beyond enumeration, we analyze the distribution of types by numb…

A New Algorithm for Computing Integer Hulls of 2D Polyhedral Sets
Chirantan Mukherjee
https://arxiv.org/abs/2509.09134 https://arxiv.org/pdf/2509.09134

A New Algorithm for Computing Integer Hulls of 2D Polyhedral Sets
The $\texttt{IntegerHull}$ function is part of Maple's $\texttt{PolyhedralSets}$ library, which calculates the integer hull of a given polyhedral set. This algorithm works by translating the supporting hyperplanes of the facets of the input polyhedral set inwards till each hyperplane encounters at least one integer point. The polyhedral set is then divided into smaller regions and a brute force method is applied to find the remaining vertices. There are certain edge case scenarios where the c…

Hyperplane arrangements and Vinberg's $\theta$-groups
Filippo Ambrosio, Andrea Santi
https://arxiv.org/abs/2509.04284 https://arxiv.org/pdf/2509.04284

Hyperplane arrangements and Vinberg's $θ$-groups
Let $\mathfrak{g} = \bigoplus_{i \in \mathbb{Z} /m \mathbb{Z}} \mathfrak{g}_i$ be a periodically graded semisimple complex Lie algebra. In this note, we give a uniform proof of the recent result by W. de Graaf and H. V. Lê that the hyperplane arrangement determined by the restrictions of the roots of $\mathfrak{g}$ to a Cartan subspace $\mathfrak{c} \subset \mathfrak{g}_1$ coincides with the hyperplane arrangement of (complex) reflections of the little Weyl group of $\mathfrak{g} = \bigoplus_{…

A Unified Optimization Framework for Multiclass Classification with Structured Hyperplane Arrangements
V\'ictor Blanco, Harshit Kothari, James Luedtke
https://arxiv.org/abs/2510.05047

A Unified Optimization Framework for Multiclass Classification with Structured Hyperplane Arrangements
In this paper, we propose a new mathematical optimization model for multiclass classification based on arrangements of hyperplanes. Our approach preserves the core support vector machine (SVM) paradigm of maximizing class separation while minimizing misclassification errors, and it is computationally more efficient than a previous formulation. We present a kernel-based extension that allows it to construct nonlinear decision boundaries. Furthermore, we show how the framework can naturally incor…

A-D-E diagrams, Hodge--Tate hyperplane sections and semisimple quantum cohomology
Sergey Galkin, Naichung Conan Leung, Changzheng Li, Rui Xiong
https://arxiv.org/abs/2509.01101 …

A-D-E diagrams, Hodge--Tate hyperplane sections and semisimple quantum cohomology
It is known that the semisimplicity of quantum cohomology implies the vanishing of off-diagonal Hodge numbers (Hodge--Tateness). We investigate which hyperplane sections of homogeneous varieties possess either of the two properties. We provide a new efficient criterion for non-semisimplicity of the small quantum cohomology ring of Fano manifolds that depends only on the Fano index and Betti numbers. We construct a bijection between Dynkin diagrams of types A, D or E, and complex Grassmannians w…

Minkowski Space holography and Radon transform
Samrat Bhowmick, Koushik Ray
https://arxiv.org/abs/2509.04974 https://arxiv.org/pdf/2509.04974

Minkowski Space holography and Radon transform
We relate a free scalar field in the Minkowski spacetime with a scalar field with a certain scaling dimension on a sphere of codimension two. This is realised by first performing a Radon transform of the ``bulk" field on the Minkowski space to a field on a hyperplane identified with a de Sitter or Euclidean anti-de Sitter slice. The Radon transform is identified in turn with a scalar field obtained from a sphere of one further lower dimension through the so-called bulk reconstruction programme.…

Rost nilpotence for twisted Milnor hypersurfaces
Charles De Clercq, Evan Marth, Kirill Zainoulline
https://arxiv.org/abs/2510.07095 https://arxiv.org/pdf/2…

Rost nilpotence for twisted Milnor hypersurfaces
We show that the strong Rost nilpotence holds for motives of generic hyperplane sections of twisted Milnor hypersurfaces. Hence, we provide a new family of examples of smooth projective algebraic varieties which satisfy the strong Rost nilpotence principle. As an application, we compute the $p$-canonical dimension for such varieties.

Normalisers of parabolic subgroups of Artin--Tits groups and Tits cone intersections
Owen Garnier, Edmund Heng, Anthony Licata, Oded Yacobi
https://arxiv.org/abs/2509.21915 http…

Normalisers of parabolic subgroups of Artin--Tits groups and Tits cone intersections
Let $Γ$ be a Coxeter diagram and let $J \subseteq Γ$. Motivated by 3-fold flops, Iyama and Wemyss study the hyperplane arrangement in the Tits cone intersection of $J$, which is a $J$-relative generalisation of the classical Coxeter arrangement. For $Γ$ of finite-type, we show that its complexified hyperplane complement is a $K(π,1)$ space for the normaliser (quotient) of the standard parabolic subgroup of the Artin--Tits group attached to $J$. For general $Γ$ we show that Brink--Howlett's…

Multiplicatively dependent integer vectors on a hyperplane
Muhammad Afifurrahman, Valentio Iverson, Gian Cordana Sanjaya
https://arxiv.org/abs/2510.10855 https://

Multiplicatively dependent integer vectors on a hyperplane
We establish several asymptotic formulae and upper bounds for the count of multiplicatively dependent integer vectors that lie on a fixed hyperplane and have bounded height. This work constitutes a direct extension of the results obtained by Pappalardi, Sha, Shparlinski, and Stewart.

White's Conjecture for Paving Matroids
Yu-Chuan Yu, Chi Ho Yuen
https://arxiv.org/abs/2510.04163 https://arxiv.org/pdf/2510.04163

White's Conjecture for Paving Matroids
White's conjecture asserts that any two tuples of matroid bases that have the same multi-set union can be transformed from one to another by symmetric exchanges; it also implies that the toric ideals of matroids are generated by the binomials encoding these exchanges. We prove White's conjecture for the class of paving matroids. Our strategy is to generalize the inductive argument using circuit-hyperplane relaxations in the recent work of Han et al. to stressed hyperplane relaxations.

Jacobian Ideals of Hyperplane Arrangements and their Graded Betti Numbers
Juan Migliore, Uwe Nagel
https://arxiv.org/abs/2508.12113 https://arxiv.org/pdf/2…

Jacobian Ideals of Hyperplane Arrangements and their Graded Betti Numbers
A hyperplane arrangement $\cA$ is said to be free if the corresponding Jacobian ideal $J_\cA$ is Cohen-Macaulay. If $\cA$ is free then $J_\cA$ is unmixed (i.e. equidimensional). Freeness is an important property, yet its presence is not well understood. A conjecture of Terao says that freeness of $\cA$ depends only on the intersection lattice of $\cA$. Given an arrangement $\cA$, we define the ideal $J_\cA^{top}$ to be the intersection of the codimension 2 primary components of $J_\cA$. This id…

Split hyperplane sections on smooth polarized $K3$-surfaces
Alex Degtyarev
https://arxiv.org/abs/2509.24349 https://arxiv.org/pdf/2509.24349

Split hyperplane sections on smooth polarized $K3$-surfaces
We study hyperplane sections of smooth polarized $K3$-surfaces that split into unions of lines. We describe the dual adjacency graphs of such sections and find sharp upper bounds on their number. In most cases (starting from degree $6$), we also compute all configurations of such sections.

Convex bodies with sections with hyperplanes of symmetry
Efr\'en Morales-Amaya
https://arxiv.org/abs/2509.17326 https://arxiv.org/pdf/2509.17326…

Convex bodies with sections with hyperplanes of symmetry
Let $K\subset \mathbb{R}^n$ be a convex body and let $p$ in the interior of $ K$, $n \geq 3$. The point $p$ is said to be a \textit{Larman point} of $K$ if, for every hyperplane $Π$ passing through $p$, the section $Π\cap K$ has a $(n-2)$-plane of symmetry. If, in addition, for every hyperplane $Π$ passing through $p$, the section $Π\cap K$ has a $(n-2)$-plane of symmetry which contains $p$, then the point $p$ is called a revolution point. In this work we prove that if for the c…

Resonance varieties and Lie algebras of matroids and hyperplane arrangements
Alexandru Suciu
https://arxiv.org/abs/2509.24060 https://arxiv.org/pdf/2509.24…

Resonance varieties and Lie algebras of matroids and hyperplane arrangements
The resonance varieties, the holonomy Lie algebra, and the holonomy Chen Lie algebra associated with the Orlik-Solomon algebra of a matroid provide an algebraic lens through which to examine the rich combinatorial structure of matroids and their geometric realizations as complex hyperplane arrangements. In this survey paper, we emphasize the commonalities between these objects but also the possible ways in which realizable and non-realizable matroids differ, leading to some open questions.

Vanishing theorems for combinatorial geometries
Christopher Eur, Alex Fink, Matt Larson
https://arxiv.org/abs/2510.05207 https://arxiv.org/pdf/2510.05207…

Vanishing theorems for combinatorial geometries
We establish strong vanishing theorems for line bundles on wonderful varieties of hyperplane arrangements, and we show that the resulting positivity properties of Euler characteristics extend to all matroids. We achieve this by showing that every degeneration of a wonderful variety within the permutohedral toric variety is reduced and Cohen--Macaulay. The same holds for a larger class of subschemes in products of projective lines that we call "kindred," which are characterized by matroidal Hilb…

A two-point phase recovering from holographic data on a single plane
Roman Novikov, Vladimir Sivkin
https://arxiv.org/abs/2509.22048 https://arxiv.org/pdf/…

A two-point phase recovering from holographic data on a single plane
We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in $\mathbb{R}^d,$ $d\geq 2$. In this region, we consider a hyperplane $X$ with sufficiently large distance $s$ from the origin in ${\mathbb R}^d.$ We give two-point local formulas for approximate recovering the radiation solution restricted to the plane $X$ from the intensity of the total solution at $X$, that is, from holographic data. The recovering…

Replaced article(s) found for math.KT. https://arxiv.org/list/math.KT/new
[1/1]:
- Line and Hyperplane GT-variants
Florian Pop, Adam Topaz
https://

The $K(\pi, 1)$ conjecture for affine Artin groups
Giovanni Paolini, Mario Salvetti
https://arxiv.org/abs/2509.00445 https://arxiv.org/pdf/2509.00445

The $K(π, 1)$ conjecture for affine Artin groups
In this summary paper, we present the key ideas behind the recent proof of the $K(π, 1)$ conjecture for affine Artin groups, which states that complements of locally finite affine hyperplane arrangements with real equations and stable under orthogonal reflections are aspherical. We survey three facets of the argument: the combinatorics of noncrossing partition posets associated with Coxeter groups; the appearance of dual Artin groups and the question of their isomorphism with standard Artin gr…

Characteristic quasi-polynomials of deletions of Shi arrangements of type C and type D
Akihiro Higashitani, Masato Konoike, Norihiro Nakashima, Satoshi Ono
https://arxiv.org/abs/2509.02043

Characteristic quasi-polynomials of deletions of Shi arrangements of type C and type D
Characteristic quasi-polynomials enumerate the number of points in the complement of hyperplane arrangements modulo positive integers. In this paper, we compute the characteristic quasi-polynomials of the restrictions of the Shi arrangements of type C and type D by one given hyperplane, respectively. The case of type C is established by extending the method developed in our previous work on type B (\cite{HN2024}), while the case of type D is deduced through a direct connection with the results …

Hadamard ranks of algebraic varieties
Dario Antolini, Guido Mont\'ufar, Alessandro Oneto
https://arxiv.org/abs/2510.05231 https://arxiv.org/pdf/2510.05…

Hadamard ranks of algebraic varieties
Motivated by the study of decompositions of tensors as Hadamard products (i.e., coefficient-wise products) of low-rank tensors, we introduce the notion of Hadamard rank of a given point with respect to a projective variety: if it exists, it is the smallest number of points in the variety such that the given point is equal to their Hadamard product. We prove that if the variety $X$ is not contained in a coordinate hyperplane or a binomial hypersurface, then the generic point has a finite $X$-Had…

Addition theorems for Ziegler pairs of hyperplane arrangements
Takuro Abe, Lukas K\"uhne, Piotr Pokora
https://arxiv.org/abs/2509.19011 https://arxiv.…

Stochastic Non-T\^atonnement Processes and the Attraction Principle
Leandro Lyra Braga Dognini
https://arxiv.org/abs/2509.17248 https://arxiv.org/pdf/2509.…

Stochastic Non-Tâtonnement Processes and the Attraction Principle
I characterize stochastic non-tâtonnement processes (SNTP) and argue that they are a natural outcome of General Equilibrium Theory. To do so, I revisit the classical demand theory to define a normalized Walrasian demand and a diffeomorphism that flattens indifference curves. These diffeomorphisms are applied on the three canonical manifolds in the consumption domain (i.e., the indifference and the offer hypersurfaces and the trade hyperplane) to analyze their images in the normalized and the f…

Taming Irrationality: An Invariance Principle for the Random Billiard Walk
Ruben Carpenter
https://arxiv.org/abs/2508.12849 https://arxiv.org/pdf/2508.1284…

Taming Irrationality: An Invariance Principle for the Random Billiard Walk
The random billiard walk is a stochastic process $(L_t)_{t\geq 0}$ in which a laser moves through the Coxeter arrangement of an affine Weyl group in $\mathbb{R}^d$, reflecting at each hyperplane with probability $p\in (0, 1)$ and transmitting unchanged otherwise. Defant, Jiradilok, and Mossel introduced this process from the perspective of algebraic combinatorics and established that, for initial directions aligned with the coroot lattice, $L_t/\sqrt{t}$ converges to a centered spherical Gaussi…

A discrete approach to Zhang's projection inequality
David Alonso-Guti\'errez, Eduardo Lucas, Javier Mart\'in Go\~ni
https://arxiv.org/abs/2509.14986 https://…

A discrete approach to Zhang's projection inequality
In this paper we will provide a new proof of the fact that for any convex body $K\subseteq\R^n$ $$ \frac{{{2n}\choose{n}}}{n^n}n\int_0^\infty r^{n-1}\vol_n(K\cap(re_n+K))dr\leq\frac{(\vol_n(K))^{n+1}}{(\vol_{n-1}(P_{e_n^\perp}(K)))^n}, $$ where $(e_i)_{i=1}^n$ denotes the canonical orthonormal basis in $\R^n$, $P_{e_n^\perp}(K)$ denotes the orthogonal projection of $K$ onto the linear hyperplane orthogonal to $e_n$, and $\vol_k$ denotes the $k$-dimensional Lebesgue measure. This inequality was …

Brauer groups of certain affine cubic surfaces
Abdulmuhsin Alfaraj
https://arxiv.org/abs/2509.16042 https://arxiv.org/pdf/2509.16042

Brauer groups of certain affine cubic surfaces
We study the Brauer groups of affine surfaces that are complements of singular hyperplane sections of smooth cubic surfaces over a field $k$ of characteristic $0$. We determine the Brauer group over the algebraic closure as a Galois module for all the possible singular hyperplane sections. For the case when the hyperplane section is geometrically the union of three lines and $k=\mathbb{Q}$, we give examples where the Galois invariant part descends to $\mathbb{Q}$.

Hamiltonian Cycles in Simplicial and Supersolvable Hyperplane Arrangements
Veronika K\"orber, Tobias Schnieders, Jan Stricker, Jasmin Walizadeh
https://arxiv.org/abs/2508.14538

Hamiltonian Cycles in Simplicial and Supersolvable Hyperplane Arrangements
Motivated by the Gray code interpretation of Hamiltonian cycles in Cayley graphs, we investigate the existence of Hamiltonian cycles in tope graphs of hyperplane arrangements, with a focus on simplicial, reflection, and supersolvable arrangements. We confirm Hamiltonicity for all 3-dimensional simplicial arrangements listed in the Grünbaum--Cuntz catalogue. Extending earlier results by Conway, Sloane, and Wilks, we prove that all restrictions of finite reflection arrangements, including all We…

Bijection Between Point-Hyperplane Anti-Flags of $V(n, 2)$ and Non-Singular Points of $O^ (2n, 2)$
Ferdinand Ihringer, Antonio Pasini
https://arxiv.org/abs/2509.14798 https://…

Bijection Between Point-Hyperplane Anti-Flags of $V(n, 2)$ and Non-Singular Points of $O^+(2n, 2)$
We give a bijection between the point-hyperplane antiflags of $V(n, 2)$ and the non-singular points of $V(2n, 2)$ with respect to a hyperbolic quadric. With the help of this bijection, we give a description of the strongly regular graph $NO^+(2n, 2)$ in $V(2n, 2)$. We also describe a graph in $V(2n, 2)$ that was recently defined by Stanley and Takeda in $V(n, 2)$.

Kinematic flow from the flow of cuts
Ross Glew, Andrzej Pokraka
https://arxiv.org/abs/2508.11568 https://arxiv.org/pdf/2508.11568

Kinematic flow from the flow of cuts
The wavefunction coefficients of conformally coupled scalars in power-law FRW cosmologies satisfy differential equations governed by a set of simple combinatorial rules known as the kinematic flow. In this paper we derive the kinematic flow, expressed using a set of differential forms referred to as the cut basis, from a geometric perspective, relying solely on the cosmological hyperplane arrangement and without invoking bulk physics. Each element of the cut basis corresponds to the positive ge…

Solomon-Terao polynomials and Castelnouvo-Mumford regularity of hyperplane arrangements
Takuro Abe
https://arxiv.org/abs/2509.10047 https://arxiv.org/pdf/2…

Solomon-Terao polynomials and Castelnouvo-Mumford regularity of hyperplane arrangements
The Solomon-Terao bi-polynomial was introduced by Solomon and Terao which degenerates to the characteristic polynomial of hyperplane arrangements. Also, it was proved recently that the other specialization of the Solomon-Terao bi-polynomial, we call the Solomon-Terao polynomial, coincides with the Poinraré polynomial of the regular nilpotent Hessenberg variety when the arrangement and the variety comes from the same lower ideal in the positive system. Moreover, there are recent developments wi…

Stable hyperplane arrangements
Toshio Oshima
https://arxiv.org/abs/2510.11099 https://arxiv.org/pdf/2510.11099

Stable hyperplane arrangements
We classify complex hyperplane arrangements $\mathcal A$ whose intersection posets $L(\mathcal A)$ satisfy $L(\mathcal A)=π_i^{-1}\circπ_i\bigl(L(\mathcal A)\bigr)$ for $i=1,\dots,n$. Here $π_i$ denotes the projection from $\mathbb C^n$ onto $\mathbb C^{n-1}$ defined by that forgets the coordinate $x_i$ of $(x_1,\dots,x_n)\in\mathbb C^n$, and $π_i\bigl(L(\mathcal A)\bigr)=\{π_i(S)\mid S\in L(\mathcal A)\}$. We show that such arrangements $\mathcal A$ arise as pullbacks of the mirror hyperp…

Crosslisted article(s) found for math.RT. https://arxiv.org/list/math.RT/new
[1/1]:
- The characteristic quasi-polynomials of hyperplane arrangements with actions of finite groups
Ryo Uchiumi

Chromatic quasisymmetric functions for signed graphs
Jean-Christophe Aval, Raquel Melgar
https://arxiv.org/abs/2508.20200 https://arxiv.org/pdf/2508.20200

Chromatic quasisymmetric functions for signed graphs
In 1995, Stanley introduced the chromatic symmetric function of a graph, which specializes to its chromatic polynomial, and which has been the focus of intense research. In 2017, Shareshian, Wachs, and Ellzey defined a refinement of this function for a directed graph, that appears to be in $QSym$, the algebra of quasisymmetric functions, which is of great interest in algebraic combinatorics. Our goal is to extend this work to signed graphs, taking into account the perspective of the hyperplane …

Reconstrucion of oriented matroids from Varchenko-Gelfand algebras
Yukino Yagi, Masahiko Yoshinaga
https://arxiv.org/abs/2509.19905 https://arxiv.org/pdf/2…

Reconstrucion of oriented matroids from Varchenko-Gelfand algebras
The algebra of $R$-valued functions on the set of chambers of a real hyperplane arrangement is called the Varchenko-Gelfand (VG) algebra. This algebra carries a natural filtration by the degree with respect to Heaviside functions, giving rise to the associated graded VG algebra. When the coefficient ring $R$ is an integral domain of characteristic $2$, the graded VG algebra is known to be isomorphic to the Orlik-Solomon algebra. In this paper, we study VG algebras over coefficient rings of ch…

Stochastically Evolving Graphs via Edit Semigroups
Fan Chung, Sawyer Jack Robertson
https://arxiv.org/abs/2509.19678 https://arxiv.org/pdf/2509.19678

Stochastically Evolving Graphs via Edit Semigroups
We investigate a randomly evolving process of subgraphs in an underlying host graph using the spectral theory of semigroups related to the Tsetlin library and hyperplane arrangements. Starting with some initial subgraph, at each iteration, we apply a randomly selected edit to the current subgraph. Such edits vary in nature from simple edits consisting of adding or deleting an edge, or compound edits which can affect several edges at once. This evolving process generates a random walk on the set…

Exponents of $2$-multiarrangements and Wakefield--Yuzvinsky matrices
Shota Maehara
https://arxiv.org/abs/2509.16569 https://arxiv.org/pdf/2509.16569…

Exponents of $2$-multiarrangements and Wakefield--Yuzvinsky matrices
In the theory of hyperplane arrangements, M. Wakefield and S. Yuzvinsky utilized a square matrix in their research on the exponents of $2$-dimensional multiarrangements. Using such a matrix, they showed that the exponents of $2$-dimensional multiarrangements are as close as possible in general position for any fixed balanced multiplicity. In this article, we introduce a matrix similar to that of Wakefield and Yuzvinsky and explore its further application to the exponents. In fact, exponents of …

To cover a permutohedron
Bochao Kong, Ji Zeng
https://arxiv.org/abs/2509.13877 https://arxiv.org/pdf/2509.13877

To cover a permutohedron
The permutohedron $P_n$ of order $n$ is a polytope embedded in $\mathbb{R}^n$ whose vertex coordinates are permutations of the first $n$ natural numbers. It is obvious that $P_n$ lies on the hyperplane $H_n$ consisting of points whose coordinates sum up to $n(n+1)/2$. We prove that if the vertices of $P_n$ are contained in the union of $m$ affine hyperplanes different from $H_n$, then $m\geq n$ when $n \geq 3$ is odd, and $m \geq n-1$ when $n \geq 4$ is even. This confirms a recent conjecture o…