arXiv subjects

Masamichi Kuroda

Publications and source records attributed to Masamichi Kuroda.

9 recordsLinked to original sources

arXiv2026

The Coboundary Quasi-Polynomials of Hyperplane Arrangements over Residually Finite Dedekind Domains

The characteristic polynomial plays an important role in study of hyperplane arrangements. There are several refinements of the characteristic polynomial. One of them is the coboundary polynomial defined by Crapo. Another refinement is the characteristic quasi-polynomial for an integral arrangement defined by Kamiya, Takemura, and Terao. Recently, the first and third authors introduced the characteristic quasi-polynomial for arrangement defined over a residually finite Dedekind domain. In this article, we introduce the common refinement of the coboundary polynomial and characteristic quasi-polynomial for an arrangement over a residually finite Dedekind domain.

math.CO↗

arXiv2026

Geometric characterization of $p$-exceptional monomial GAPN functions

On finite fields of characteristic $p$, PN (perfect nonlinear) functions for odd $p$ and APN (almost perfect nonlinear) functions for even $p$ are well-known classes of highly nonlinear functions. GAPN (generalized almost perfect nonlinear) functions were introduced as a generalization of APN functions for even $p$ to all $p$. One of the main targets of studies on such highly nonlinear functions is their classification. While $p$-exceptional monomial PN and $2$-exceptional monomial APN functions have been classified, the corresponding problem for GAPN functions remains open. Here, a polynomial over $\mathbb{F}_p$ is called a $p$-exceptional PN (resp. APN, resp. GAPN) function if it is a PN (resp. APN, resp. GAPN) function on $\mathbb{F}_{p^n}$ for infinitely many positive integers $n$. In this paper, we give a geometric characterization of $p$-exceptional monomial GAPN functions. To this end, for a prime power $q$, we establish a geometric characterization of non-zero polynomials in $\mathbb{F}_{q}[x,y]$ having no absolutely irreducible factor defined over $\mathbb{F}_{q}$.

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arXiv2023

The Characteristic Quasi-Polynomials of Hyperplane Arrangements over Residually Finite Dedekind Domains

Kamiya, Takemura, and Terao initiated the theory of the characteristic quasi-polynomial of an integral arrangement, which is a function counting the elements in the complement of the arrangement modulo positive integers. They gave a period of the characteristic quasi-polynomial, called the LCM-period, and showed that the first constituent of the characteristic quasi-polynomial coincides with the characteristic polynomial of the corresponding hyperplane arrangement. Recently, Liu, Tran, and Yoshinaga showed that the last constituent of the characteristic quasi-polynomial coincides with the characteristic polynomial of the corresponding toric arrangement. In addition, by using the theory of toric arrangements, Higashitani, Tran, and Yoshinaga proved that the LCM-period is the minimum period of the characteristic quasi-polynomial. In this paper, we study an arrangements over a Dedekind domain such that every residue ring with a nonzero ideal is finite and give algebraic generalizations of the above results.

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arXiv2021

Chromatic Signed-Symmetric Functions of Signed Graphs

Stanley introduced the chromatic symmetric function of a simple graph, which is a generalization of a chromatic polynomial. This is expressed in terms of the integer points of the complements of the corresponding graphic arrangement. Stanley proved a combinatorial reciprocity theorem for chromatic functions. This is considered as an Ehrhart-type reciprocity theorem for the graphic arrangement. We introduce the chromatic signed-symmetric function of a signed graph, an analogue of the chromatic symmetric function, by the integer points of the complements of the corresponding signed-graphic arrangement and prove a generalization of Stanley's reciprocity theorem. Stanley has conjectured that the chromatic symmetric function distinguishes trees. This conjecture is also generalized for signed trees. We verify the conjecture for certain classes of signed paths.

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arXiv2020

Unit Ball Graphs on Geodesic Spaces

Consider finitely many points in a geodesic space. If the distance of two points is less than a fixed threshold, then we regard these two points as "near". Connecting near points with edges, we obtain a simple graph on the points, which is called a unit ball graph. If the space is the real line, then it is known as a unit interval graph. Unit ball graphs on a geodesic space describe geometric characteristics of the space in terms of graphs. In this article, we show that every unit ball graph on a geodesic space is (strongly) chordal if and only if the space is an $ \mathbb{R} $-tree and that every unit ball graph on a geodesic space is (claw, net)-free if and only if the space is a connected manifold of dimension at most $ 1 $. As a corollary, we prove that the collection of unit ball graphs essentially characterizes the real line and the unit circle.

math.CO↗

arXiv2017

Monomial generalized almost perfect nonlinear functions

Generalized almost perfect nonlinear (GAPN) functions were defined to satisfy some generalizations of basic properties of almost perfect nonlinear (APN) functions for even characteristic. In this paper, we study monomial GAPN functions for odd characteristic. In particular, we give all monomial GAPN functions whose algebraic degree are maximum or minimum on a finite field of odd characteristic.

math.CO↗

arXiv2017

The GIT moduli of semistable pairs consisting of a cubic curve and a line on ${\mathbb P}^{2}$

We discuss the GIT moduli of semistable pairs consisting of a cubic curve and a line on the projective plane. We study in some detail this moduli and compare it with another moduli suggested by Alexeev. It is the moduli of pairs (with no specified semi-abelian action) consisting of a cubic curve with at worst nodal singularities and a line which does not pass through singular points of the cubic curve. Meanwhile, we make a comparison between Nakamura's compactification of the moduli of level three elliptic curves and these two moduli spaces.

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arXiv2016

A Generalization of APN Functions for Odd Characteristic

Almost perfect nonlinear (APN) functions on finite fields of characteristic two have been studied by many researchers. Such functions have useful properties and applications in cryptography, finite geometries and so on. However APN functions on finite fields of odd characteristic do not satisfy desired properties. In this paper, we modify the definition of APN function in the case of odd characteristic, and study its properties.

math.CO↗

arXiv2013

$\mathbb Q$-bases of the Néron-Severi groups of certain elliptic surfaces

P. Stiller computed the Picard numbers of several families of elliptic surfaces, the rank of the Néron-Severi groups of these surfaces. However he did not give the generators of these groups. In this paper we give $\mathbb Q$-bases of these groups explicitly. If these surfaces are rational, then we also show that they are $\mathbb Z$-bases.

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