1998 Fiscal Year Final Research Report Summary
Foundation of computational Commutative algebra with a view toward combinatorics on convex polytopes
| Project/Area Number |
09440013
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Osaka University |
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Principal Investigator |
HIBI Takayuki Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80181113)
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| Co-Investigator(Kenkyū-buntansha) |
YANAGAWA Koji Graduate School of Science, Assistant, 大学院・理学研究科, 助手 (40283006)
NAMIKAWA Yoshinori Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (80228080)
MIYANISHI Masayoshi Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80025311)
SUZUKI Takashi Graduate School of Science, Professor, 大学院・理学研究科, 教授 (40114516)
KAWANAKA Noriaki Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10028219)
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| Project Period (FY) |
1997 – 1998
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| Keywords | componentwise linear / generic initial ideal / Cohen-Macaulay ring / squaretree inonomial / lexsegment ideal / graded Betti number / polynomial ring / Kruskal-Katona thearem |
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| Research Abstract |
The important activity during the period of the present research project is, first, to present the concept of componentwise linear ideals and to establish its fundamental theory and, second, to study generic initial ideals of simplicial complexes and to discuss their concrete and effective applications to combinatorics. First of all, we obtained the theorem that the squarefree monomial ideal associated with a simplicial complex is componentwise linear if and only if its dual complex is sequentially Cohen-Macaulay, and explained the algebraic aspect of sequentially Cohen-Macaulay complexes and their h-triangles. Second, based on fundamental study about generic initial ideals of coruponentwise linear ideals, the important result that a homogeneous ideal of the polynomial ring possesses the stable Betti numbers if and only if the ideal is componentwise linear was established. Such the theorem guarantees that componentwise linear ideals will play an important role in computational commutative algebra. Third, in order to obtain sophisticated generalization of Kruskal-Katona theorem in classical combinatorics on finite sets, via the discussion on the existence of a squarefree strongly stable ideal having the same graded Betti numbers as those of the generic initial ideal of a squarefree ideal in the polynomial ring, we did succeed in obtaining the affirmative answer to the outstanding conjecture that the graded Betti numbers of a squarefree ideal with a fixed Hubert function are less than or equal to those of the lexsegment ideal.
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