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離散幾何的な新概念とグレブナー基底理論の融合による凸多面体論における新手法の開発

Research Project

Project/Area Number 18H01134
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Review Section Basic Section 12030:Basic mathematics-related
Research InstitutionKwansei Gakuin University

Principal Investigator

大杉 英史  関西学院大学, 理学部, 教授 (80350289)

Co-Investigator(Kenkyū-buntansha) 東谷 章弘  大阪大学, 情報科学研究科, 准教授 (60723385)
Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Granted (Fiscal Year 2022)
Budget Amount *help
¥16,120,000 (Direct Cost: ¥12,400,000、Indirect Cost: ¥3,720,000)
Fiscal Year 2022: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2021: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2020: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2019: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2018: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Keywordsグレブナー基底 / 凸多面体 / エルハート多項式
Outline of Annual Research Achievements

凸多面体に含まれる格子点の数え上げや三角形分割は離散幾何学に止まらず様々な分野に現れる基本的かつ重要な研究対象である。当該研究の目的は,近年発明された離散幾何における革新的な新手法とグレブナー基底理論の融合・相互作用により,グレブナー基底理論に基づく新手法の開発を進めることである。
今年度の主たる研究成果としては,以下の3つがあげられる。
1.有限グラフに付随するPQ型隣接凸多面体と呼ばれる格子多面体について研究し,ジョイングラフを考えた場合の,PQ型隣接凸多面体のh*多項式に関する公式を与えた。この公式を応用し,車輪グラフのPQ型隣接凸多面体の正規化体積の明示的な公式を与えることにも成功し,Chen-Davisの予想を肯定的に解決した。また,完全多部グラフについてもh*多項式や正規化体積の公式を与え,Chen-Davisの完全2部グラフ K_{2,m} に関する結果を大幅に拡張することができた。
2.自然数の分割に付随するシュペヒトイデアルと呼ばれるイデアルについて研究し,シュペヒトイデアルに関する Haiman-Wooの結果(シュペヒトイデアルの被約性,および,普遍グレブナー基底の構成)の簡潔な別証明を与えた。
3.シュペヒトイデアルのステイト多面体が一般化されたpermutohedron であることを証明した。この結果によって特に,与えられたシュペヒトイデアルが持つイニシャルイデアルの個数を,分割の大きさ n と,連続する成分の差の最小値 k を用いた式 n!/(k+1)!で表すことができた。

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

イデアルのグレブナー基底を完全に把握する「ステイト多面体」を求めることは通常非常に困難であり,シュペヒトイデアルに関する結果は貴重である。活用したアイデアを他の重要なクラスにも拡張できる可能性がある。

Strategy for Future Research Activity

正規多面体,反射的凸多面体などの重要な性質を持つ整凸多面体について,既存の著名なクラスの解析や,新しい豊富なクラスの発見に努める。これらの研究成果をもとに,理論的な考察に加えて,計算機支援によって豊富なクラスによる実証を遂行し,離散幾何における最新の理論をグレブナー基底を始めとする可換代数的手法と融合し,関連する理論の発展を目指す。

Report

(4 results)
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report

Research Products

(35 results)

All 2022 2021 2020 2019 2018 Other

All Journal Article (13 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 13 results,  Open Access: 8 results) Presentation (19 results) (of which Int'l Joint Research: 7 results,  Invited: 7 results) Book (1 results) Remarks (2 results)

  • [Journal Article] Edge rings with q-linear resolutions2022

    • Author(s)
      Mori Kenta、Ohsugi Hidefumi、Tsuchiya Akiyoshi
    • Journal Title

      Journal of Algebra

      Volume: 593 Pages: 550-567

    • DOI

      10.1016/j.jalgebra.2021.11.018

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Quadratic Groebner bases of block diagonal matching field ideals and toric degenerations of Grassmannians2022

    • Author(s)
      Higashitani Akihiro、Ohsugi Hidefumi
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 226

    • DOI

      10.1016/j.jpaa.2021.106821

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Symmetric edge polytopes and matching generating polynomials2021

    • Author(s)
      Ohsugi Hidefumi、Tsuchiya Akiyoshi
    • Journal Title

      Combinatorial Theory

      Volume: 1

    • DOI

      10.5070/c61055371

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] The h*-Polynomials of Locally Anti-Blocking Lattice Polytopes and Their γ-Positivity2021

    • Author(s)
      Ohsugi Hidefumi、Tsuchiya Akiyoshi
    • Journal Title

      Discrete & Computational Geometry

      Volume: 66 Pages: 701-722

    • DOI

      10.1007/s00454-020-00236-6

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Enriched order polytopes and enriched Hibi rings2021

    • Author(s)
      Hidefumi Ohsugi, Akiyoshi Tsuchiya
    • Journal Title

      European Journal of Mathematics

      Volume: published online Pages: 48-68

    • DOI

      10.1007/s40879-020-00403-2

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Enriched chain polytopes2020

    • Author(s)
      Ohsugi Hidefumi、Tsuchiya Akiyoshi
    • Journal Title

      Israel Journal of Mathematics

      Volume: - Pages: 485-500

    • DOI

      10.1007/s11856-020-2012-1

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Reflexive polytopes arising from bipartite graphs with $$\gamma $$-positivity associated to interior polynomials2020

    • Author(s)
      Ohsugi Hidefumi、Tsuchiya Akiyoshi
    • Journal Title

      Selecta Mathematica

      Volume: 26 Pages: 59-59

    • DOI

      10.1007/s00029-020-00588-0

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Integer decomposition property for Cayley sums of order and stable set polytopes2020

    • Author(s)
      Takayuki Hibi, Hidefumi Ohsugi, Akiyoshi Tsuchiya
    • Journal Title

      Michigan Mathematical Journal

      Volume: 掲載確定 Pages: 765-778

    • DOI

      10.1307/mmj/1585792887

    • Related Report
      2019 Annual Research Report 2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Toric ideals of Minkowski sums of unit simplices2020

    • Author(s)
      Akihiro Higashitani, Hidefumi Ohsugi
    • Journal Title

      Algebraic Combinatorics

      Volume: 掲載確定

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Nef-partitions arising from unimodular configurations2020

    • Author(s)
      Hidefumi Ohsugi, Akiyoshi Tsuchiya
    • Journal Title

      Mathematische Nachrichten

      Volume: 掲載確定

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Toric rings and ideals of stable set polytopes2019

    • Author(s)
      Kazunori Matsuda, Hidefumi Ohsugi, Kazuki Shibata
    • Journal Title

      Mathematics

      Volume: 7 Pages: 613-613

    • DOI

      10.3390/math7070613

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Arithmetic aspects of symmetric edge polytopes2019

    • Author(s)
      Akihiro Higashitani, Katharina Jochemko, Mateusz Michalek
    • Journal Title

      Mathematika

      Volume: 掲載確定

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Smooth centrally symmetric polytopes in dimension 3 are IDP2019

    • Author(s)
      Matthias Beck, Christian Haase, Akihiro Higashitani, Johannes Hofscheier, Katharina Jochemko, Lukas Katthaen, Mateusz Michalek
    • Journal Title

      Annals of Combinatorics

      Volume: 掲載確定

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Quadratic Groebner bases of block matching field ideals2021

    • Author(s)
      大杉英史
    • Organizer
      MFO-RIMS Tandem Workshop “Symmetries on polynomial ideals and varieties”
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Edge rings with q-linear resolutions2021

    • Author(s)
      毛利健太
    • Organizer
      第17回数学総合若手研究集会
    • Related Report
      2020 Annual Research Report
  • [Presentation] Block diagonal matching field イデアルとグラスマン多様体のトーリック退化2021

    • Author(s)
      東谷章弘、大杉英史
    • Organizer
      日本数学会年会
    • Related Report
      2020 Annual Research Report
  • [Presentation] Edge rings with q-linear resolutions2021

    • Author(s)
      毛利健太、大杉英史、土谷昭善
    • Organizer
      日本数学会年会
    • Related Report
      2020 Annual Research Report
  • [Presentation] 対称辺凸多面体とマッチング生成多項式2021

    • Author(s)
      大杉英史、土谷昭善
    • Organizer
      日本数学会年会
    • Related Report
      2020 Annual Research Report
  • [Presentation] 単位単体のミンコフスキ和のトーリックイデアル2020

    • Author(s)
      大杉英史
    • Organizer
      組合せ論と可換環論
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Quadratic Groebner bases of block diagonal matching field ideals and toric degenerations of Grassmannians2020

    • Author(s)
      大杉英史
    • Organizer
      東京可換環論セミナー
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Unimodular 配置に付随する nef 分割2020

    • Author(s)
      大杉英史、土谷昭善
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2020 Annual Research Report
  • [Presentation] Two enriched poset polytopes2019

    • Author(s)
      土谷昭善
    • Organizer
      Toric Topology 2019 in Okayama
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Locally anti-blocking lattice polytopes and their h*-polynomials2019

    • Author(s)
      土谷昭善
    • Organizer
      Commutative Algebra and Lattice Polytopes
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Enriched Hibi ring2019

    • Author(s)
      大杉英史,土谷昭善
    • Organizer
      日本数学会2019年度秋季総合分科会
    • Related Report
      2019 Annual Research Report
  • [Presentation] Two enriched poset polytopes2019

    • Author(s)
      大杉英史,土谷昭善
    • Organizer
      日本数学会2019年度秋季総合分科会
    • Related Report
      2019 Annual Research Report
  • [Presentation] Ehrhart theory and interior polynomials2019

    • Author(s)
      土谷昭善
    • Organizer
      Recent advances in matroids and Tutte polynomials
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Reflexive polytopes arising from bipartite graphs with γ-positivity associated to interior polynomials2019

    • Author(s)
      Hidefumi Ohsugi, Akiyoshi Tsuchiya
    • Organizer
      AMS Spring Central and Western Sectional Meeting
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 順序凸多面体と安定集合凸多面体のケーリー和とその正規性およびGorenstein性2019

    • Author(s)
      大杉英史,土谷昭善,日比孝之
    • Organizer
      日本数学会年会
    • Related Report
      2018 Annual Research Report
  • [Presentation] 二部グラフに付随するある反射的凸多面体のγ-positive 性と内部多項式の関係2019

    • Author(s)
      大杉英史,土谷昭善
    • Organizer
      日本数学会年会
    • Related Report
      2018 Annual Research Report
  • [Presentation] Regular unimodular triangulations of dilated empty simplices and Grobner basis2018

    • Author(s)
      Akihiro Higashitani
    • Organizer
      JCCA 2018
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Characterization problem on Ehrhart polynomials of lattice polytopes2018

    • Author(s)
      Akihiro Higashitani
    • Organizer
      AlgebraGeometryCombinatorics Seminar
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 有限グラフに付随する反射的凸多面体のδ列について2018

    • Author(s)
      東谷章弘
    • Organizer
      応用数学合同研究集会
    • Related Report
      2018 Annual Research Report
  • [Book] Binomial Ideals2018

    • Author(s)
      Juergen Herzog, Takayuki Hibi, Hidefumi Ohsugi
    • Total Pages
      321
    • Publisher
      Springer International Publishing
    • ISBN
      9783319953472
    • Related Report
      2018 Annual Research Report
  • [Remarks] 関西学院大学理学部大杉英史

    • URL

      https://sci-tech.ksc.kwansei.ac.jp/~hohsugi/index.html

    • Related Report
      2021 Annual Research Report 2020 Annual Research Report
  • [Remarks] 関西学院大学理工学部大杉英史

    • URL

      https://sci-tech.ksc.kwansei.ac.jp/~hohsugi/index.html

    • Related Report
      2019 Annual Research Report

URL: 

Published: 2018-04-23   Modified: 2022-12-28  

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