Making use of algebraic and combinatorial techniques, we study two topics: the arithmetic degree of squarefree strongly stable ideals and the h-vectors of matroid complexes. For a squarefree monomial ideal, I, the arithmetic degree of I is the number of facets of the simplicial complex which has I as its Stanley-Reisner ideal. We consider the case when I is squarefree strongly stable, in which case we give an exact formula for the arithmetic degree in terms of the minimal generators of I as well as a lower bound resembling that from the Multiplicity Conjecture. Using this, we can produce an upper bound on the number of minimal generators of any Cohen-Macaulay ideals with arbitrary codimension extending Dubreil’s theorem for codimension 2. A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible h-vectors of such complexes. In the case when the complex has dimension 1 we completely resolve this question and we give some partial results for higher dimensions. We also prove the 1-dimensional case of a conjecture of Stanley that all matroid h-vectors are pure O-sequences. Finally, we completely characterize the Stanley-Reisner ideals of matroid complexes.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23676号 / 理博第4766号 / 新制||理||1683(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 森脇 淳, 教授 雪江 明彦, 教授 吉川 謙一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21532号 / 理博第4439号 / 新制||理||1638(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 伊藤 哲史, 教授 雪江 明彦, 教授 池田 保 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

<p>Let S be a polynomial ring K[x1,...,xn] over a field K and let F be a non-negatively graded free module over S generated by m basis elements. In this thesis, we study four kinds of upper bounds: degree bounds for Gröbner bases of submodules of F, bounds for arithmetic degrees of S-ideals, regularity bounds for radicals of S-ideals, and Stillman bounds. </p> <p><br></p> <p>Let M be a submodule of F generated by elements with degrees bounded above by D and dim(F/M)=r. We prove that if M is graded, the degree of the reduced Gröbner basis of M for any term order is bounded above by 2[1/2((Dm)^{n-r}m+D)]^{2^{r-1}}. If M is not graded, the bound is 2[1/2((Dm)^{(n-r)^2}m+D)]^{2^{r}}. This is a generalization of bounds for ideals in a polynomial ring due to Dubé (1990) and Mayr-Ritscher (2013).</p> <p><br></p> <p>Our next results are concerned with a homogeneous ideal I in S generated by forms of degree at most d with dim(S/I)=r. In Chapter 4, we show how to derive from a result of Hoa (2008) an upper bound for the regularity of sqrt{I}, which denotes the radical of I. More specifically we show that reg(sqrt{I})<= d^{(n-1)2^{r-1}}. In Chapter 5, we show that the i-th arithmetic degree of I is bounded above by 2*d^{2^{n-i-1}}. This is done by proving upper bounds for arithmetic degrees of strongly stable ideals and ideals of Borel type.</p> <p><br></p> <p>In the last chapter, we explain our progress in attempting to make Stillman bounds explicit. Ananyan and Hochster (2020) were the first to show the existence of Stillman bounds. Together with G. Caviglia, we observe that a possible way of making their results explicit is to find an effective bound for an invariant called D(k,d) and supplement it into their proof. Although we are able to obtain this bound D(k,d) and realize Stillman bounds via an algorithm, it turns out that the computational complexity of Ananyan and Hochster's inductive proof would make the bounds too large to be meaningful. We explain the bad behavior of these Stillman bounds by giving estimates up to degree 3.</p>