자료유형  

 학위논문

Control Number  

0016932792

International Standard Book Number  

9798379844486

Dewey Decimal Classification Number  

541.395

Main Entry-Personal Name  

Liang, Yihui.

Publication, Distribution, etc. (Imprint  

[S.l.] : Purdue University., 2022

Publication, Distribution, etc. (Imprint  

Ann Arbor : ProQuest Dissertations & Theses, 2022

Physical Description  

1 online resource(73 p.)

General Note  

Source: Dissertations Abstracts International, Volume: 85-01, Section: B.

General Note  

Advisor: Caviglia, Giulio.

Dissertation Note  

Thesis (Ph.D.)--Purdue University, 2022.

Restrictions on Access Note  

This item must not be sold to any third party vendors.

Summary, Etc.  

요약Let S be a polynomial ring K[x1, . . . , xn] over a field K and let Fbe 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 Gr¨obner bases of submodules of F, bounds for arithmetic degrees of S-ideals, regularity bounds for radicals of S-ideals, and Stillman bounds.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 Gr¨obner basis of M for any term order is bounded above by 2 [1/2((Dm) n−rm + D)]2 r−1 . If M is not graded, the bound is 2 h 1/2((Dm) (n−r) 2m + D) i2 r. This is a generalization of bounds for ideals in a polynomial ring due to Dub´e (1990) and Mayr-Ritscher (2013).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 √ I, which denotes the radical of I. More specifically we show that reg(√ I) ≤ d (n−1)2r−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.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.

Subject Added Entry-Topical Term  

Homogenization.

Subject Added Entry-Topical Term  

Codes.

Subject Added Entry-Topical Term  

Algebra.

Subject Added Entry-Topical Term  

Mathematics.

Added Entry-Corporate Name  

Purdue University.

Host Item Entry  

Dissertations Abstracts International. 85-01B.

Host Item Entry  

Dissertation Abstract International

Electronic Location and Access  

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Control Number  

joongbu:641397