2022 Fiscal Year Final Research Report
On the unimodality of delta polynomials of normal lattice polytopes
| Project/Area Number |
19K14505
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Toho University (2022)
The University of Tokyo (2019-2021) |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2023-03-31
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| Keywords | 格子凸多面体 / δ多項式 / γ非負性 / 反射的凸多面体 / 正規凸多面体 |
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| Outline of Final Research Achievements |
The aim of this project is to investigate properties of the delta-polynomials, which are related to counting lattice points, of normal reflexive polytopes. In particular, we focus on gamma-positivity, which is a property between unimodality and real-rootedness. Our results of this project are (1) construction of symmetric edge polytopes of type B and a study on their delta-polynomials, (2) construction of enriched order polytopes and enriched chain polytopes, and a study on their delta-polynomials, (3) a study on the delta-polynomials of locally antiblocking polytopes, (4) a study on the delta-polynomials of symmetric edge polytopes. In (1) and (2), we, in particular, proved the gamma-positivity of the delta-polynomials, and in (3) and (4), for a large class of the polytopes, we proved the gamma-positivity and give a conjecture on the gamma-positivity.
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| Free Research Field |
代数的組合せ論
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| Academic Significance and Societal Importance of the Research Achievements |
数え上げに関連する多項式や数列の研究は古典的な組合せ論の問題意識であるが，特に多項式が回文的，つまり係数の列が対称，となるときは単峰性より強い性質であるγ非負性の研究が最近注目を集めている．格子凸多面体のδ多項式のγ非負性に関する研究はこれまでほとんどなく，今回の研究において，多くのγ非負なδ多項式を持つ格子凸多面体を構成でき，今後のγ非負性に関する研究の礎ができた．またいくつかの予想を提唱したところ，国内外で取り組む研究者が出てくるなど，格子凸多面体論の研究の方向性を与えることに成功し，今後の発展が期待できる．
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