arXiv subjects

Ryoshun Oba

Publications and source records attributed to Ryoshun Oba.

7 recordsLinked to original sources

arXiv2026

Frobenius identities for the volume map on Cohen--Macaulay rings

We study the volume map on Artinian quotients of Cohen-Macaulay algebras in characteristic $p$, and the interaction between it and the action of Frobenius on resolutions. This allows us to provide a general, conceptual way to understand Parseval-Rayleigh identities, curious inhomogeneous identities on the volume map which were developed for the proof of the Ohsugi-Hibi conjecture. This general perspective gives a new approach to generic Lefschetz theory. We use this perspective to do the following: we give sufficient conditions for anisotropy and the Hard Lefschetz property for generic Artinian reductions of graded Gorenstein rings; we study the codimension-$3$ Gorenstein quotient of a polynomial ring by the ideal generated by Pfaffians, proving a Parseval-Rayleigh identity and deriving anisotropy and Hard Lefschetz in characteristic $2$; we deduce the $g$-theorem for simplicial spheres and the Ohsugi-Hibi conjecture following previous work of Adiprasito, Papadakis, and Petrotou; and we provide further examples of Parseval-Rayleigh identities for Gorenstein rings.

math.AC↗

arXiv2025

Parseval-Rayleigh identities for homogeneous complete intersections

We prove, in any positive characteristic, Parseval-Rayleigh identities for the residue map of a homogeneous complete intersection. As an application, we give a conceptual proof of the folklore fact that generic homogeneous complete intersections have the Strong Lefschetz Property over any field of characteristic 2.

math.AC↗

arXiv2024

Multigraded strong Lefschetz property for balanced simplicial complexes

Generalizing the strong Lefschetz property for an $\mathbb{N}$-graded algebra, we introduce the multigraded strong Lefschetz property for an $\mathbb{N}^m$-graded algebra. We show that, for $\mathbf{a} \in \mathbb{N}^m_+$, the generic $\mathbb{N}^m$-graded Artinian reduction of the Stanley-Reisner ring of an $\mathbf{a}$-balanced homology sphere over a field of characteristic $2$ satisfies the multigraded strong Lefschetz property. A corollary is the inequality $h_{\mathbf{b}} \leq h_{\mathbf{c}}$ for $\mathbf{b} \leq \mathbf{c} \leq \mathbf{a}-\mathbf{b}$ among the flag $h$-numbers of an $\mathbf{a}$-balanced simplicial sphere. This can be seen as a common generalization of the unimodality of the $h$-vector of a simplicial sphere by Adiprasito and the balanced generalized lower bound inequality by Juhnke-Kubitzke and Murai. We further generalize these results to $\mathbf{a}$-balanced homology manifolds and $\mathbf{a}$-balanced simplicial cycles over a field of characteristic $2$.

math.CO↗

arXiv2023

Rigidity of Balanced Minimal Cycle Complexes

A $(d-1)$-dimensional simplicial complex $\Delta$ is balanced if its graph $G(\Delta)$ is $d$-colorable. Klee and Novik obtained the balanced lower bound theorem for balanced normal $(d-1)$-pseudomanifolds $\Delta$ with $d\geq3$ by showing that the subgraph of $G(\Delta)$ induced by the vertices colored in $T$ is rigid in $\mathbb{R}^3$ for any $3$ colors $T$. We show that the same rigidity result, and thus the balanced lower bound theorem, holds for balanced minimal $(d-1)$-cycle complexes with $d \geq 3$. Motivated by the Stanley's work on a colored system of parameters for the Stanley-Reisner ring of balanced simplicial complexes, we further investigate the infinitesimal rigidity of non-generic realization of balanced, and more broadly $\bm{a}$-balanced, simplicial complexes. Among other results, we show that for $d \geq 4$, a balanced homology $(d-1)$-manifold can be realized as an infinitesimally rigid framework in $\mathbb{R}^d$ such that each vertex of color $i$ lies on the $i$th coordinate axis.

math.CO↗

arXiv2023

Realizable Dimension of Periodic Frameworks

Belk and Connelly introduced the realizable dimension $\textrm{rd}(G)$ of a finite graph $G$, which is the minimum nonnegative integer $d$ such that every framework $(G,p)$ in any dimension admits a framework in $\mathbb{R}^d$ with the same edge lengths. They characterized finite graphs with realizable dimension at most $1$, $2$, or $3$ in terms of forbidden minors. In this paper, we consider periodic frameworks and extend the notion to $\mathbb{Z}$-symmetric graphs. We give a forbidden minor characterization of $\mathbb{Z}$-symmetric graphs with realizable dimension at most $1$ or $2$, and show that the characterization can be checked in polynomial time for given quotient $\mathbb{Z}$-labelled graphs.

math.CO↗

arXiv2022

Super Stable Tensegrities and the Colin de Verdi\`{e}re Number $\nu$

A super stable tensegrity introduced by Connelly in 1982 is a globally rigid discrete structure made from stiff bars or struts connected by cables with tension. In this paper we show an exact relation between the maximum dimension that a multigraph can be realized as a super stable tensegrity and Colin de Verdi\`{e}re number~$\nu$ from spectral graph theory. As a corollary we obtain a combinatorial characterization of multigraphs that can be realized as 3-dimensional super stable tensegrities.

math.CO↗

arXiv2020

Characterizing the Universal Rigidity of Generic Tensegrities

A tensegrity is a structure made from cables, struts and stiff bars. A $d$-dimensional tensegirty is universally rigid if it is rigid in any dimension $d'$ with $d'\geq d$. The celebrated super stability condition due to Connelly gives a sufficient condition for a tensegrity to be universally rigid. Gortler and Thurston showed that super stability characterizes universal rigidity when the point configuration is generic and every member is a stiff bar. We extend this result in two directions. We first show that a generic universally rigid tensegrity is super stable. We then extend it to tensegrities with point group symmetry, and show that this characterization still holds as long as a tensegrity is generic modulo symmetry. Our strategy is based on the block-diagonalization technique for symmetric semidefinite programming problems, and our proof relies on the theory of real irreducible representation of finite groups.

math.OC↗