Grant amount：\4290000 （ Direct expense: \3300000 、 Indirect expense：\990000 ）

We give a calculation method of the diagonal F-thresholds on binomial hypersurfaces. Moreover, we prove an inequality on the diagonal F-thresholds, the a-invariant (due to Goto and Watanabe) and the F-pure thresholds on standard graded affine toric rings. We discuss Cohen-Macaulay properties of powers of edge ideals of simple graphs, and characterize graphs higher powers of edge ideals for which are Cohen-Macaulay. Furthermore, we give a similar result in the case of second power. As an application of Skoda's theorem, we prove a variant of Wang's theorem (about Goto number).

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Grant amount：\4160000 （ Direct expense: \3200000 、 Indirect expense：\960000 ）

The minimal free resolution of a projective variety is one of the most important algebraic invariants measuring the complexity of the defining ideal of the variety. My research focuses on bounding the Castelnuovo-Mumford regularity introduced by Mumford. Upper bounds on the regularity of Buchsbaum varieties is known to be described in terms of the degree and the codimension of the variety. I have obtained that a Buchsbaum variety with extremal and next-extremal case having Castelnuovo-type regularity bound is a divisor on either a variety of minimal degree or Del Pezzo variety. Also I have obtained a Horrocks-type criterion on the splitting of vector bundles on multiprojective space as an application of Castelnuovo-Mumford regularity.

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Grant amount：\4420000 （ Direct expense: \3400000 、 Indirect expense：\1020000 ）

We studied Stanly-Reisner ideals, which are squarefree monomial ideals in polynomial rings. As a result we have proved that any powers areCohen-Macaulay if a certain m-th power of a Stanley-Reisner ideal is Cohen-Macaulay, where m is more than two. In this case the original ideal is a complete intersection. This is a refinement of the Cowsik-Nori theorem in the case of monomial ideals.

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Grant amount：\4420000 （ Direct expense: \3400000 、 Indirect expense：\1020000 ）

On the core subjects of this research theme ; Number Theory, specifically Hasse's problem related to Abelian fields[A], Arithmetic, Algebraic geometry[B] and its application to Discrete Mathematics[C], we held the Workshop on Number Theory in Saga in each August and January of 2008~2010. The research organizer stayed at NUCES during two and half years to work the joint research with PhD scholars at both campuses. On Hasse's problem, the organizer obtained the characterization on the monogeneity for certain family of pure sextic and pure octic fields by the joint work with PhD scholars at NUCES. In our research on algebraic geometry codes, Cooperative Uehara gave an idea of constructing a new class of algebraic geometry codes different from the known codes of one-point type. Also, applying the concept of evaluation codes, which generalize one-point-type algebraic geometry codes, Uehara invented a method of constructing codes from integer rings on algebraic number fields, and presented some explicit examples of such codes[C]. Cooperative Miyazaki studied the minimal free resolution of Buchsbaum varieties and obtained a classification of the Buchsbaum variety in terms of the Castelnuovo-Mumford regularity[B]. Cooperative Terai studied Stanly-Reisner ideals, which are squarefree monomial ideals in polynomial rings[B]. Cooperative Katayama . has determined finite symplectic groups of cube and 4th order, using the structure of the unit groups of cubic and quadratic fields, and announced these results at the workshop in Saga 2011. Newman, Shanks and Williams determined finitesymplectic groups of square order in1980's. Katayama also investigated the number of congruent k-polygons inscribed in a unit circle, where the vertices chosen from n division points of the circle[C]. Cooperative Taguchi studied the ramification theory of truncated discrete valuation rings (= : tdvr's) and the (non)existence of mod p Galois representations. On the former, Taguchi proved (jointly with T. Hiranouchi) that the category of finite extensions of a tdvr A is equivalent to a category of finite extensions, with restricted ramification, of a complete discrete valuation field which lifts A. On the latter, Taguchi proved (jointly with H. Moon) the non-existence of 2-dimensional mod 2 Galois representations for some quadratic fields[B].

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Grant amount：\4550000 （ Direct expense: \3500000 、 Indirect expense：\1050000 ）

By studying arithmetic geometry of Siegel modular varieties, we solved the congruence problem of Siegel modular forms, and showed that weights of p-adicSiegel modular forms are determined as p-adic numbers. Further, we constructed a basictheory of arithmetic vector-valued Siegel modular forms and vector-valued p-adic Siegel modular forms with natural p-adic operators. Moreover, we studied the ring structure ofSiegel modular forms over rings in which 6 is invertible, and decided this structure in the degree 2 case.

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Grant amount：\8450000 （ Direct expense: \6500000 、 Indirect expense：\1950000 ）

The leader of this research introduced the notion of a generalized tight closure and succeeded to redefine multiplier ideals in terms of commutative ring theory. Precisely speaking, the multiplier ideal is given as the limit of generalized test ideals. In this research, we found several differences between the behavior of test ideals and that of multiplier ideals. Moreover, we extended the theory of test ideals in order to study several invariants in the theory of commutative algebra.

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Grant amount：\3900000 （ Direct expense: \3000000 、 Indirect expense：\900000 ）

The author has been applied the theory of derived categories (e.g., Koszul duality) to the study of combinatorial commutative algebra. The main aim of this research project was to extend this idea and results to general commutative algebra. In this direction, results related to the coherent property of rings were obtained, and the paper was published in an academic journal in 2009. Around 2008, the author slightly changed the direction of the research, and begun to study relatively new objects of combinatorial commutative algebra for which we cannot use usual methods. In this direction, the author has written several papers. Some of them have been published in an academic journal.

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Grant amount：\3880000 （ Direct expense: \3400000 、 Indirect expense：\480000 ）

A07) Investigation of Hasse's problem for the power integral bases and Its Application
On Hasse's problem, the head investigator, S. I. A. Shah(NUCES), Y. Motoda
(Yatsushiro National College of Tech.) and Uehara gave a new family of infinitely many monogenic cyclic quartic fields using a linear combination among partial differences.
We investigated the Diophantine equations related to the binary recurrence sequences. We also investigated its application to the construction of an infinite family of cyclic extensions of degree p-1 having the p-ranks of the class groups of at least two[Katayama].
B07) Applications of number theory to arithmetic geometry and algebraic geometry
we extended results of Swinnerton-Dyer, Serre and Katz on congruence and p-adic properties of elliptic modular forms [Ichikawa].We studied (jointly with Hyunsuk Moon) the 1-adic properties of certain modular forms and proved the non-existence and finiteness of mod 2 Galois representations of some quadratic fields [Taguchi].
C07) Applications of number theory to coding theory and discrete mathematics
We have researched on a new method of constructing Hermitian codes which are error-correcting codes constructed from Hermitian curves. we have investigated into finding an explicit expression of a basic of the trace-norm code [Uehara]. We studied Stanley-Reisner rings which are locally complete intersections. As a result, we proved that locally complete intersection Stanley-Reisner rings are complete intersections if the corresponding simplicial complexes are of dimension more than 1 and are connected [Terai].

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Grant amount：\4010000 （ Direct expense: \3500000 、 Indirect expense：\510000 ）

1. We showed that any stable vector bundle of degree 0 on a Riemann surface close to a maximally degenerate curve is obtained from a linear representation of the Schottky group uniformizing the Riemann surface. Further, we described the relationship between the representation space of the Schottky group and the moduli space of the vector bundles by using Abel-Jacobi's theorem and Verlinde's formula.
2. We described the ring structure of Siegel modular forms of degree 2 over a ring containing 1/6 extending Igusa's result. Further, we extended results of Swinnerton-Dyer, Serre and Katz on congruence and p-adic properties of elliptic modular forms to the case of Siegel modular forms.
3. We gave a mathematical rigorous model of the one loop approximation of the perturbative Chern-Simons integral in an abstract Wiener space setting, and by appealing to the Malliavin-Taniguchi formula of changing variables on the abstract Wiener space, we derived the asymptotic expansion for the Chern-Simons integral with respect to the charge parameter.

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Grant amount：\4010000 （ Direct expense: \3500000 、 Indirect expense：\510000 ）

The purpose of this research is to study algebraic and combinatorial properties of minimal free resolution of Stanley-Reisner rings. We focused on the relation between the multiplicity and the Castelnuovo-Mumford regularity of Stanley-Reisner rings.
Before the academic year 2005 we proved that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to the dimension d of the Stanley-Reisner rings if its multiplicity is less than or equal to d. Moreover we verified that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to d if its multiplicity is less than or equal to 2d-1, and if the degree of generators of the Stanley-Reisner ideal is less than or equal to d
In the academic year 2006, developing these results, we proved that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to d if its multiplicity is less than or equal to 3d-2, and if the degree of generators of the first syzygy module of the Stanley-Reisner ideal is less than or equal to d+1. From this result we conjectured that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to d if its multiplicity is less than or equal to (p+2)d- (p-1), and if the degree of generators of the p-th syzygy module of the Stanley-Reisner ideal is less than or equal to d+p. In the academic year 2007 we proved that the above conjecture holds if the dimension of the Stanley-Reisner rings is 2 or 3. We also found that this conjecture is a generalization of the lower bound theorem, that is famous in convex polytope theory, in the facet case.

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Grant amount：\3450000 （ Direct expense: \3000000 、 Indirect expense：\450000 ）

1. We have researched on Hermitian codes which are constructed by Hermitian curves, and have found a new method of constructing of them. By our method we have constructed new Hermitian codes different from known ones, that is, one of the one-point type, and we have further computed a lower bound for their minimum distances. As a result, we have proved that our Hermitian codes have better properties than ones of the one-point type.
2. We carried out an investigation on finding an explicit linear basis of the trace-norm code, and have found its explicit form when the number of variables is less than or equal to 3.
3. We have researched on algebraic construction of low density parity check codes, and have shown methods of constructions of them by vector spaces over a finite field and by non-abelian groups.
4. We have studied on structure of integral bases of algebraic number fields, and have proved that there is only one case that the integer ring of a 2-elementary abelian field of degree greater than or equal to 8 is generated by a single integer.
5. We have researched on Siegel modular forms, and have described the ring structure of Siegel modular forms of degree 2 over a ring containing 1/6.
6. We have studied Stanley-Reisner rings which are locally complete intersections. As a result, we proved that locally complete intersection Stanley-Reisner rings are complete intersections if the corresponding simplicial complexes are of dimension more than 1 and are connected.

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Grant amount：\3880000 （ Direct expense: \3400000 、 Indirect expense：\480000 ）

射影空間内において有限個の斉次多項式の零点として定義された射影多様体の定義イデアルの次数、極小自由分解の複雑さを表す重要な不変量として、Castelnuovo-Mumford 量がある。Castelnuovo-Mumford 量の上限を射影多様体の次数、余次元などで記述する問題はこの分野の重要なテーマであり、いくつかの上限が知られている。本研究において、射影曲線のCastelnuovo-Mumford 量がCastelnuovo 型の上限、次の上限を満たすときに、最小次数の射影曲面もしくは正規Del Pezzo 曲面の因子となることを示した。

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Grant amount：\3600000 （ Direct expense: \3600000 ）

The purpose of this research is to study algebraic and combinatorial properties of minimal free resolution of Stanley-Reisner rings and to consider its combinatorial applications.
In the academic year 2004 we studied Buchsbaum Stanley-Reisner rings with linear resolution. We determined the lower bound for the multiplicity of Stanley-Reisner rings. And we showed that they have linear resolution if they possess the minimal multiplicity. We also showed a necessary and sufficient condition for Buchsbaum Stanley-Reisner rings to have linear resolution in terms of the reduced homology groups of the corresponding simplicial complex and its links.
In the academic year 2005 we mainly studied the relation between the multiplicity of Stanley-Reisner rings and their Castelnuovo-Mumford regularity. We proved that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to the dimension d if its multiplicity is less than or equal to d. Moreover we verified that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to d if its multiplicity is less than or equal to 2d-1, and if the degree of generators of the Stanley-Reisner ideal is less than or equal to d
We also investigated Stanley-Reisner rings with d-linear resolution among those with linear resolution intensively. Using the above result we showed that a Stanley-Reisner ring has d-linear resolution if its multiplicity is less than or equal to d and if the degree of generators of the Stanley-Reisner ideal is more than or equal to d. Moreover we showed that a Stanley-Reisner ring has d-linear resolution if its multiplicity is less than or equal to 2d-1 and if the degree of generators of the Stanley-Reisner ideal is d. By Alexander duality, we also verified that a Stanley-Reisner ring is Cohen-Macaulay if its multiplicity is large enough.

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Grant amount：\8600000 （ Direct expense: \8600000 ）

The lattice ideal of dimension zero appears in the research of both pure mathematics and applied mathematics. The original purpose of the present research project was to establish the algebraic theory of universal Groebner bases of lattice ideal of dimension zero and to study of its theoretical effectivity to commutative algebra and algebraic geometry as well as its practical effectivity to integer programming, coding theory together with algebraic statistics. First, we investigated the universal Groebner basis of the lattice ideal of dimension zero arising from the toric ideal of a finite graph and succeeded in describing its structure in terms of the finite graph. Second, in the study of a problem on integer programming arising from a finite graph for which Gomory's relaxation can be applied, we developed the technique to decide the estimation of the computational complexity of finding an optimal solution by using combinatorics on finite graphs. Third, we achieved the study of finding an explicit expression of the corner polyhedron of a lattice ideal of dimension zero in terms of the Minkowski sum of a bounded convex polytope and a convex cone, and obtained some results on the combinatorics of the polyhedral structure of the corner polyhedron. Finally, we developed the algebraic study on the Markov basis of the contingency table in algebraic statistics and presented a statistic model arising from a complete multipartite graph. These research results will contribute to the development of the algebraic study of integer programming. In addition, we organized two international meetings related with computational commutative algebra and Groebner bases.

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Grant amount：\3700000 （ Direct expense: \3700000 ）

(1)Using arithmetic Schottky-Mumford uniformization theory on algebraic curves, we constructed Teichmuller groupoids in the category of arithmetic geometry. By this construction, we gave a partial answer to Grothendieck's conjecture on the associated Galois representations, and described monodromy representations induced from confbrmal field theory.
(2)Extending Ullmo-Zhang's results on Bogomolov's conjecture, we gave a condition that a subvariety of an abelian variety defined over a number field is isomorphic to an abelian variety in terms of Neron-Tate's height functions.
(3)We described the structure of Riemann surfaces defined from the monodromy representation of hypergeometric differential equations with purely imaginary exponents (joint work with M.Yoshida).
(4)We determined the structure of the class groups and unit groups of algebraic number fields of Kummer type, specifically of quartic Dirichlet fields (joint work with K.Katayama and C.Levesque). Further, we investigated the problem of Hasse concerning power integral basis of the ring of algebraic integers (joint work with Y.Motoda).
(5)We defined stochastic holonomy operator and the Chern-Simons integral of some product of gauge invariant Wilson loop observables in the Wiener space setting.
(6)We studied Buchsbaum Stanley-Reisner rings with linear resolution and characterized them by their multiplicity. Further, we studied the arithmetical rank and determined it for monomial ideals of deviation two.
(7)We studied 4-manifolds having flexible surfaces inside, and showed that a lot of simply connected 4-manifolds not the 4-sphere have flexible surfaces. Further, we introduced an operation to alter any surface in a simply connected 4-manifold into a flexible surface.

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Grant amount：\2100000 （ Direct expense: \2100000 ）

当該企画調査では,Oberwoltach型の中規模準備会議を開催し,当該分野の研究の動向を詳細に分析し,研究目的に列挙した研究領域(トーリックイデアルとグレブナー基底,整数計画とGomory relaxation,0次元ラティスイデアルの普遍グレブナー基底,単項式イデアルの極小自由分解,幾何学的なBuchbergerアルゴリズムの高速化)の妥当性を慎重に審議した。その準備会議の概要を列挙する。[1]可換代数におけるアルゴリズム的手法(責任者:寺井直樹/於:大阪大学/平成15年7月)斉次代数の極小自由分解とベッチ数列,トーリック環の正則度と重複度などを題材とし,可換代数におけるアルゴリズム的手法を議論した。[2]有限グラフと0次元ラティスイデアル(責任者:大杉英史/於:立教大学/平成15年11月)有限グラフの隣接行列から生起する0次元ラティスイデアルを可換代数と組合せ論の両面から具象的に探求し,未解決問題を集約した。[3]グレブナー基底と応用数学(責任者:大杉英史/於:立教大学/平成16年1月)整数計画における代数的手法の有効性,Gomory relaxationと算術次数,符号理論と統計数学におけるトーリックイデアルとグレブナー基底の有効性について研究した。[4]可換代数と代数幾何(責任者:日比隆之/於:大阪大学/平成16年3月)いわゆるaffine algebraic geometryとその周辺領域,多項式環の組合せ論についての国際会議である。海外からの参加者はJurgen Herzogら7名であった。

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Grant amount：\3500000 （ Direct expense: \3500000 ）

We performed the. research of algebraic geometry codes which are error-correcting codes constructed from algebraic function fields, and related researches in algebraic number theory, arithmetic algebraic geometry, algebraic geometry and algebraic combinatrics. The aim of this project is to construct algebraic geometry codes explicitly applying algebraic function fields and to determine their minimum distances, which are.important numbers for estimating their abilities of correcting errors.
In the research of algebraic geometry codes, we determined the minimum distance d(C) of certain type of algebraic geometry codes C, called one-point algebraic geometry codes, in the first academic year. Speaking in detail, we proved that the minimum distance d(C) of a one-point algebraic geometry code C is equal to its Feng-Rao lower bound d' (C) if C'satisfies some conditions. In the second, academic year, we construct algebraic geometry codes other than of one-point type, and computed their Feng -Rao lower bounds. As a result, we found some algebraic geometry codes whose Feng-Rao lower bound are larger than the corresponding codes of-one-point type.
As a research in algebraic number theory, we investigated the class number and the structure of the unit groups for algebraic number fields of lower extension degree over the rationals, specifically for quartic number fields of Kummer extension. Also we concerned ourselves with the question whether the integer ring of an abelian field of degree 8 hasa power basis.
As a research in arithmetic algebraic geometry, we constructed the Teichmueller groupoids in the category of arithmetic geometry, and we described the Galois action and the monodromy representation (associated with conformal field theory) on the Teichmueller groupoids. Furthermore we proved the Bogomolov conjecture which states that if an irreducible curve in an abelian variety is not, isomorphic to an elliptic curve, then its algebraic points are distributed uniformly discretely for the Neron-Tate height.
As a research in algebraic geometry, we considered the problem to estimate the degree of the Chow variety oil-cycles of degree d in the n-th projective space, and investigated a connection between resultants, which are projective invariants, and some Hilbert polynomials.
As a reaearch in algebraic combinatrics, we investigated a minimal free resolution of the Stanley-Reisnerring of a simplicial complex. In particular, we give an upper bound on the dimension of the Unique non-vanishing homology group of a Buchsbaum Stanley-Reisner ring with linear resolution.

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Grant amount：\2000000 （ Direct expense: \2000000 ）

現在研究代表者らは国際研究集会「凸多面体を巡る組合せ数学の代数的諸相」を平成16年7月に札幌で開催する準備を進めているが,根幹となる研究領域を(1)グレブナー基底と組合せ数学(2)凸多面体の三角形分割と整数計画(3)外積代数とalgebraic shifting(4)単項式イデアルの極小自由分解(5)斉次整域の正則度と重複度,とする原案が有力である.当該企画調査では,当該分野の昨今の研究動向などに関する周到な調査を遂行し,上記項目を研究集会の研究領域の根幹とすることの妥当性を吟味するため,Oberwolfach型の中規模国内準備会議を2回開催した.すなわち,「グレブナー基底の理論的有効性と実践的有効性」(責任者:大杉英史/於:京都大学/平成14年7月)と「ジェネリックイニシャルイデアルの研究」(責任者:寺井直樹/於:大阪大学/平成14年12月)である.前者においては凸多面体の三角形分割と整数計画問題,符号理論と暗号理論などにおけるグレブナー基底の果たす役割について多角的に研究した.後者においては多項式環と外積代数のジェネリックイニシャルイデアルの相互関係を可換代数と組合せ論の両面から具象的に探究した.その他,平成14年6月にイタリアで開催された研究集会「可換代数の昨今の潮流」に研究代表者と研究分担者の一部が参加し,当該分野の研究の進展状況を把握した.当該企画調査の結論として,上記の根幹となる研究領域はすべて妥当であると判断され,個々の研究領域において当該国際研究集会に相応しい話題を選別する作業を推進した.

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Grant amount：\3900000 （ Direct expense: \3900000 ）

1. We described the monodromy representation of Teichmueller goupoids associated with conformal field theory. Extending Ullmo-Zhang's result on the Bogomolov conjecture, we gave a condition that a subvariety of an abelian variety is isomorphic to an abelian variety in terms of the value distribution of a Neron-Tate height function on the subvariety. We described the Riemann surfaces associated with the monodromy representation of hypergeometric equation with purely imaginary exponents.
2. We gave an explicit formula of the Hasse unit index for the unit group of quadratic fields, and considered a Problem of Hasse for the ring of integers in certain abelian fields.
3. We tried to justify the perturbative Chern-Simons theory using the asymptotic expansion theory via infinite dimensional stochastic analysis, and derived a simple Homfly polynomial.
4. We showed that for certain algebraic geometry codes, the minimum distance are equal to the Fang-Rao bound, and found an algebraic geometry code of other type with same property.
5. We gave the upper bound for the average number of connected components of the induced subgraphs of the graphs for simplicial polytopes, and proved that the arithmetical rank is equal to the projective dimension for the almost complete intersection Stanley-Reisner ideals.
6. We calculated the virtual cohomological dimension and the Euler number of the mapping class group of a three-dimensional handlebody.

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Grant amount：\1800000 （ Direct expense: \1800000 ）

本研究調査の目的は、スタンレーライスナー環のBetti数についてその可換環論的、組合せ論的性質を考察することであり、そこからグラフ理論・組合せ論的応用を導くことであった。
本年度は、スタンレーライスナー環の極小自由分解の中でも特に2線形部分と呼ばれる部分のBetti数について重点を置いて研究した。Betti数に対する基本的な問題は、他の環論的不変量でその上限を評価することである。
報告者は、単体的凸多面体に付随するスタンレーライスナー環の2線形部分のBetti数の上限をその凸多面体の次元と頂点数を用いて与えた。また、上の評価において、ちょうど上限を与えるものはスタック多面体に付随するスタンレーライスナー環であることも示した。
一方、純で強連結な単体的複体に付随するスタンレーライスナー環の2線形部分のBetti数についてもその上限をその単体的複体の次元と頂点数を用いて与えた。また、上の評価において、ちょうど上限を与えるものは高次元木に付随するスタンレーライスナー環であることも示した。
グラフ理論における応用として、誘導部分グラフの平均連結成分数という概念を導入し、対応するスタンレーライスナー環の2線形部分のBetti数との関係を研究した。そして、その結果として、単体的凸多面体の辺グラフ及び、純で強連結な単体的複体の1骨格について誘導部分グラフの平均連結成分数の上限を与えた。

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Grant amount：\2000000 （ Direct expense: \2000000 ）

本研究調査の目的は、Stanley-Reisner環のBetti数についてその可換環論的、組合せ論的性質を考察し、また、一般の次数つき環のBetti数との関係を探ることにあった。本年度も昨年度に引き続きBetti数を研究する上で重要な不変量であるregurarityについて重点を置いて研究した。
このregurarityに対して、次のニつの概念を用いて研究した。一つはGroebner基底である。Groebner基底は、計算機における数式処理において基本的な役割を果してきた。それは、連立方程式の解などを実用のレベルで求めるのに画期的なアルゴリズムを与えているし、さらに最近は図形処理にも大きな役割を果たしている。したがって、環のヒルベルト関数を求めることは、近年、Groebner基底の理論を用いることにより、計算機によって、計算できる様になってきた。ここでは、generic Groebner基底の理論を用いて、一般の次数つき環とスタンレーライスナー環の間の不変量の関係を調べた。
もう一つは、アレクサンダー双対複体である。Eagon氏とReiner氏によって、アクレサンダー双対複体を用いて、スタンレーライスナー環におけるCohen-Macaulay性と線形な極小自由分解をもつということの間の関係が明らかにされたのが、その出発点であった。本研究においては、それをさらに一般化し、スタンレーライスナー環のdepthとそのアレクサンダー双対複体のスタンレーライスナー環のregularityの間の関係を定式化した。そのことにより、regularityの問題をdepthの問題に帰着させて研究することが可能になった。そのことを利用して、Eisenbud-Goto予想のモノミアル版について肯定的に解いた。

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Grant amount：\1800000 （ Direct expense: \1800000 ）

本研究の課題である代数幾何符号については、Feng-Raoによって単項式列による構成法が提唱されているが、この方法に従ってある種の代数曲線から生成される誤り訂正符号(代数幾何符号)を構成し、その最小距離の下からの評価を行った。これは、Feng-Raoの結果を補正し部分的に精密化したものである。この研究結果は国際シンポジウムで発表し、論文は報告集に掲載予定である。さらに、単項式列による構成法を用いて、一般の代数的集合より代数幾何符号を構成することを試み、その際の単項式列の特定方法を開発し、大きい最小距離を得るための単項式列の並べ方について実験を行った。さらにエルミート符号の最小距離の特定に関して簡単な証明法を発見した。また、代数幾何符号に対するFeng-Raoの設定最小距離と行変換と多数決原理を用いた復号法の理論が一般の線形符号にも適用できることを示した。これらの研究結果は、目下論文として整理するため準備中である。さらに、2進L関数の整数点での値に関する合同関係式の一般化に関する研究を行い、国際シンポジウムで発表した。
一方、研究分担者の研究課題については次の成果があった。1)実2次体の類群の3部分群の構造を決定した。2)ソリトン(KdV,KP)方程式の普遍テ-タ関数解及びp進テ-タ関数解を構成した。3)アレクサンドロフ空間上のラプラシアンの理論を確立した。4)テ-タ関数を用いて3次元射影空間に埋め込まれた正規楕円4次曲線のチャウ形式をテ-タ定数を用いて具体的に表した。5)スタンレーライスナ-環の極小自由分解に現れるベッチ数の組合せ論における応用について研究した。これらの研究は、殆どが口頭発表を行い論文がそれぞれの雑誌に掲載済みか掲載予定である。

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Grant amount：\1000000 （ Direct expense: \1000000 ）

多項式環を、square-freeな単項式たちによって生成されるイデアルで割った環は、Stanley-Reisner環と呼ばれる。この環は、環論的手法からだけでなく、トポロジカルな、あるいは、組合せ論な手法を用いて研究され、その環論的性質が、組合せ論にも様々に応用されてきている。Buchsbaum Stanley-Reisner環に現われるh-vectorの特徴づけに関する問題も位相多様体の三角形分割に関連して興味深い。これについて、h-vectorであるための一つの必要条件を与えた。また、低次元の場合には、その十分性についても考察した。
与えられた加群に対して、その極小自由分解を構成し、Betti数列を調べることは、大切な問題である。というのは、それは、Cohen-Macaulay性、Gorenstein性等の重要な環論的情報を含んでいるからである。しかし、Betti数列を計算することは難しく、Hilbert関数からわかる例以外はほとんど知られていなかった。そこで、巡回多面体およびstacked多面体に付随するStanley-Reisner環のBetti数列を具体的に与えた。
また、Betti数列がStanley-Reisner環の係数体に依存するかどうかは、それを組合せ論的な公式で表そうとするさい重要な問題となる。そこで、Stanley-Reisner環の第2Betti数が係数体に依存しないことを示した。これは、Bruns-Herzogによって、最初に環論的に示されたが、Alexander双対定理を用いてトポロジカルな短い証明を与えた。
一般には第3Betti数は係数体に依存するのであるが、イデアルが次数2の単項式で生成されているならば、第3及び第4Betti数も係数体に依存しないことを示した。

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