arXiv subjects

Masahiko Yoshinaga

Publications and source records attributed to Masahiko Yoshinaga.

At least 19 recordsLinked to original sources

arXiv2026

Approximate Gromov--Hausdorff continuity of magnitude and weighting

Magnitude gives an effective size of a finite metric space and is now used in several data-analysis settings. For such applications, it is natural to ask how magnitude behaves under Gromov-Hausdorff perturbations, including collisions of points. However, magnitude is nowhere continuous on finite metric spaces with the Gromov--Hausdorff topology. We show that this failure is nongeneric in a precise measure-theoretic sense. After fixing the number of points in each collapsing cluster, magnitude is approximately continuous. More strongly, when clusters of points collapse to the points of a limit space, the total weighting of each cluster approximately converges to the weighting of the corresponding limit point. The main tool is a general result on approximate limits of inverses near singular matrices.

math.MG↗

arXiv2026

Milnor fibrations and oriented matroids

We introduce a combinatorial model for the Milnor fibration of a complexified real arrangement using oriented matroids. It is a poset quasi-fibration, whose domain is a subdivision of the Salvetti complex stemming from a natural subdivision of the dual oriented matroid complex. This yields a concrete finite regular CW complex which is homotopy equivalent to the Milnor fiber of the complexified real arrangement and implies that the homotopy type of the Milnor fiber of a complexified real arrangement only depends on the underlying combinatorial structure given by its oriented matroid. Moreover, our construction works for any oriented matroid, disregarding realizability, so we obtain a notion of a combinatorial Milnor fibration for any oriented matroid.

math.AG↗

arXiv2026

Reconstruction 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 characteristic different from $2$, and investigate to what extent VG algebras determine the underlying oriented matroid structures. Our main results concern hyperplane arrangements that are generic in codimension $2$. For such arrangements, if $R$ is an integral domain of characteristic not equal to $2$, then the oriented matroid can be recovered from both the filtered and the graded VG algebras. As a byproduct, we prove that, unlike the complexification, the cohomology ring of the complement of a $3$-plexification of a real arrangement is not determined by the intersection lattice. We also formulate an algorithm that is expected to reconstruct oriented matroids from VG algebras in the case of general arrangements.

math.CO↗

arXiv2025

$q$-deformation of chromatic polynomials and graphical arrangements

We first observe a mysterious similarity between the braid arrangement and the arrangement of all hyperplanes in a vector space over the finite field $\mathbb{F}_q$. These two arrangements are defined by the determinants of the Vandermonde and the Moore matrix, respectively. These two matrices are transformed to each other by replacing a natural number $n$ with $q^n$ ($q$-deformation). In this paper, we introduce the notion of ``$q$-deformation of graphical arrangements'' as certain subarrangements of the arrangement of all hyperplanes over $\mathbb{F}_q$. This new class of arrangements extends the relationship between the Vandermonde and Moore matrices to graphical arrangements. We show that many invariants of the ``$q$-deformation'' behave as ``$q$-deformation'' of invariants of the graphical arrangements. Such invariants include the characteristic (chromatic) polynomial, the Stirling number of the second kind, freeness, exponents, basis of logarithmic vector fields, etc.

math.CO↗

arXiv2025

Maximal Betti number for local system cohomology of hyperplane arrangement complements

Let $\mathcal{L}$ be a rank one local system with field coefficient on the complement $M(\mathcal{A})$ of an essential complex hyperplane arrangement $\mathcal{A}$ in $\mathbb{C}^\ell$. Dimca-Papadima and Randell independently showed that $M(\mathcal{A})$ is homotopy equivalent to a minimal CW-complex. It implies that $\dim H^k(M(\mathcal{A}),\mathcal{L}) \leq b_k(M(\mathcal{A}))$. In this paper, we show that if $\mathcal{A}$ is real, then the inequality holds as equality for some $0\leq k\leq \ell$ if and only if $\mathcal{L}$ is the constant sheaf. The proof is using the descriptions of local system cohomology of $M(\mathcal{A})$ in terms of chambers.

math.AG↗

arXiv2025

The small-scale limit of magnitude and the one-point property

The magnitude of a metric space is a real-valued function whose parameter controls the scale of the metric. A metric space is said to have the one-point property if its magnitude converges to 1 as the space is scaled down to a point. Not every finite metric space has the one-point property: to date, exactly one example has been found of a finite space for which the property fails. Understanding the failure of the one-point property is of interest in clarifying the interpretation of magnitude and its stability with respect to the Gromov--Hausdorff topology. We prove that the one-point property holds generically for finite metric spaces, but that when it fails, the failure can be arbitrarily bad: the small-scale limit of magnitude can take arbitrary real values greater than 1.

math.MG↗

arXiv2025

Is magnitude 'generically continuous' for finite metric spaces?

Magnitude is a real-valued invariant of metric spaces which, in the finite setting, can be understood as recording the 'effective number of points' in a space as the scale of the metric varies. Motivated by applications in topological data analysis, this paper investigates the stability of magnitude: its continuity properties with respect to the Gromov-Hausdorff topology. We show that magnitude is nowhere continuous on the Gromov-Hausdorff space of finite metric spaces. Yet, we find evidence to suggest that it may be 'generically continuous', in the sense that generic Gromov-Hausdorff limits are preserved by magnitude. We make the case that, in fact, 'generic stability' is what matters for applicability.

math.MG↗

arXiv2024

Ehrhart quasi-polynomials and parallel translations

Given a rational polytope $P \subset \mathbb R^d$, the numerical function counting lattice points in the integral dilations of $P$ is known to become a quasi-polynomial, called the Ehrhart quasi-polynomial $\mathrm{ehr}_P$ of $P$. In this paper we study the following problem: Given a rational $d$-polytope $P \subset \mathbb R^d$, is there a nice way to know Ehrhart quasi-polynomials of translated polytopes $P+ \mathbf v$ for all $\mathbf v \in \mathbb Q^d$? We provide a way to compute such Ehrhart quasi-polynomials using a certain toric arrangement and lattice point counting functions of translated cones of $P$. This method allows us to visualize how constituent polynomials of $\mathrm{ehr}_{P+\mathbf v}$ change in the torus $\mathbb R^d/\mathbb Z^d$. We also prove that information of $\mathrm{ehr}_{P+\mathbf v}$ for all $\mathbf v \in \mathbb Q^d$ determines the rational $d$-polytope $P \subset \mathbb R^d$ up to translations by integer vectors, and characterize all rational $d$-polytopes $P \subset \mathbb R^d$ such that $\mathrm{ehr}_{P+\mathbf v}$ is symmetric for all $\mathbf v \in \mathbb Q^d$.

math.CO↗

arXiv2024

Magnitude homology and homotopy type of metric fibrations

In this article, we show that each two metric fibrations with a common base and a common fiber have isomorphic magnitude homology, and even more, the same magnitude homotopy type. That can be considered as a generalization of a fact proved by T. Leinster that the magnitude of a metric fibration with finitely many points is a product of those of the base and the fiber. We also show that the definition of the magnitude homotopy type due to the second and the third authors is equivalent to the geometric realization of Hepworth and Willerton's pointed simplicial set.

math.AT↗

arXiv2024

The quasi-polynomiality of mod q permutation representation for a linear finite group action on a lattice

For given linear action of a finite group on a lattice and a positive integer q, we prove that the mod q permutation representation is a quasi-polynomial in q. Additionally, we establish several results that can be considered as mod q-analogues of results by Stapledon for equivariant Ehrhart quasi-polynomials. We also prove a reciprocity-type result for multiplicities of irreducible decompositions.

math.CO↗

arXiv2024

Topology of hyperplane arrangements via real structure

This note is a survey on the topology of hyperplane arrangements. We mainly focus on the relationship between topology and the real structure, such as adjacent relations of chambers and stratifications related to real structures.

math.GT↗

arXiv2024

A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connection applied to the primitive vector field. This generalizes and unifies analogous results for real reflection groups.

math.DG↗

arXiv2024

A construction of homotopically non-trivial embedded spheres for hyperplane arrangements

We introduce the notion of locally consistent system of half-spaces for a real hyperplane arrangement. We embed a sphere in the complexified complement by shifting the real unit sphere into the imaginary direction indicated by the half-spaces. We then prove that the sphere is homotopically trivial if and only if the system of half-spaces is globally consistent. To prove its non-triviality, we compute the twisted intersection number of the sphere with a specific, explicitly constructed twisted Borel-Moore cycle.

math.GT↗

arXiv2024

Betti numbers and torsions in homology groups of double coverings

Papadima and Suciu proved an inequality between the ranks of the cohomology groups of the Aomoto complex with finite field coefficients and the twisted cohomology groups, and conjectured that they are actually equal for certain cases associated with the Milnor fiber of the arrangement. Recently, an arrangement (the icosidodecahedral arrangement) with the following two peculiar properties was found: (i) the strict version of Papadima-Suciu's inequality holds, and (ii) the first integral homology of the Milnor fiber has a non-trivial $2$-torsion. In this paper, we investigate the relationship between these two properties for double covering spaces. We prove that (i) and (ii) are actually equivalent.

math.GT↗

arXiv2024

$q$-deformation of Aomoto complex

A degree one element of the Orlik-Solomon algebra of a hyperplane arrangement defines a cochain complex known as the Aomoto complex. The Aomoto complex can be considerd as the ``linear approximation'' of the twisted cochain complex with coefficients in a complex rank one local system. In this paper, we discuss $q$-deformations of the Aomoto complex. The $q$-deformation is defined by replacing the entries of representation matrices of the coboundary maps with their $q$-analogues. While the resulting maps do not generally define cochain complexes, for certain special basis derived from real structures, the $q$-deformation becomes again a cochain complex. Moreover, it exhibits universality in the sense that any specialization of $q$ to a complex number yields the cochain complex computing the corresponding local system cohomology group.

math.GT↗

arXiv2023

Integral expressions for derivations of multiarrangements

The construction of an explicit basis for a free multiarrangement is not easy in general. Inspired by the integral expressions for quasi-invariants of quantum Calogero-Moser systems, we present integral expressions for specific bases of certain multiarrangements. Our construction covers the cases of three lines in dimension $2$ (previously examined by Wakamiko) and free multiarrangements associated with complex reflection groups (Hoge, Mano, Röhrle, Stump). Furthermore, we propose a conjectural basis for the module of logarithmic vector fields of the extended Catalan arrangement of type $B_2$.

math.CO↗

arXiv2023

Ehrhart quasi-polynomials of almost integral polytopes

A lattice polytope translated by a rational vector is called an almost integral polytope. In this paper we investigate Ehrhart quasi-polynomials of almost integral polytopes. We study the relationship between the shape of the polytopes and algebraic properties of the Ehrhart quasi-polynomials. In particular, we prove that lattice zonotopes and centrally symmetric lattice polytopes are characterized by Ehrhart quasi-polynomials of their rational translations.

math.CO↗

arXiv2023

Causal order complex and magnitude homotopy type of metric spaces

In this paper, we construct a pointed CW complex called the magnitude homotopy type for a given metric space $X$ and a real parameter $\ell \geq 0$. This space is roughly consisting of all paths of length $\ell$ and has the reduced homology group that is isomorphic to the magnitude homology group of $X$. To construct the magnitude homotopy type, we consider the poset structure on the spacetime $X\times\mathbb{R}$ defined by causal (time- or light-like) relations. The magnitude homotopy type is defined as the quotient of the order complex of an intervals on $X\times\mathbb{R}$ by a certain subcomplex. The magnitude homotopy type gives a covariant functor from the category of metric spaces with $1$-Lipschitz maps to the category of pointed topological spaces. The magnitude homotopy type also has a ``path integral'' like expression for certain metric spaces. By applying discrete Morse theory to the magnitude homotopy type, we obtain a new proof of the Mayer-Vietoris type theorem and several new results including the invariance of the magnitude under sycamore twist of finite metric spaces.

math.AT↗