京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18771号 / 理博第4029号 / 新制||理||1580(附属図書館) / 31722 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 長谷川 真人, 教授 玉川 安騎男, 准教授 照井 一成 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

My PhD-project has two main research directions. The first direction is on partial regularities which we define as refinements of the Castelnuovo-Mumford regularity. Main results are: relationship of partial regularities and related invariants, like the a-invariants or the Castelnuovo-Mumford regularity of the syzygy modules; algebraic properties of partial regularities via a filter-regular sequence or a short exact sequence; generalizing a well-known result for the Castelnuovo-Mumford regularity to the case of partial regularities of stable and squarefree stable monomial ideals; finally extending an upper bound proven by Caviglia-Sbarra to partial regularities. The second direction of my project is to develop a theory on monomial preorders. Many interesting statements from the classical theory of monomial orders generalize to monomial preorders. Main results are: a characterization of monomial preorders by real matrices, which extends a result of Robbiano on monomial orders; secondly, leading term ideals with respect to monomial preorders can be studied via flat deformations of the given ideal; finally, comparing invariants of the given ideal and the leading term ideal with respect to a monomial preorder.