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研究生: 李承修
論文名稱: Groebner Bases和Corner Elements的應用:計算Colon Ideals
An Application of Groebner Bases and Corner Elements : Computing Colon Ideals
指導教授: 劉容真
學位類別: 碩士
Master
系所名稱: 數學系
Department of Mathematics
論文出版年: 2013
畢業學年度: 101
語文別: 中文
論文頁數: 70
中文關鍵詞: 計算colon ideals
論文種類: 學術論文
相關次數: 點閱:114下載:12
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  • For every even integer n=2k, let q_n be the ideal <x^{2n},y^{2n},(xy+z^2)^n,z^n> in the polynomial ring R=Q[x,y,z]. In her master's thesis [Y], Yao gives a Groebner basis G for q_n and proves that q_n+I_n contain in (q_n:m), where m is the maximal ideal <x,y,z> of R and I_n is the monomial ideal (x^k)(y^k)(z^{2k-1})<x^{2k},y^{2k}><x,y>^{k-1} of R. In this thesis, we prove that (q_n:m) and q_n+I_n are indeed equal. In the process of proving this equality, we give a Groebner basis for the ideals q_n+I_n and find the corner elements of the monomial ideal <LM(q_n)>.

    For every even integer n=2k, let q_n be the ideal <x^{2n},y^{2n},(xy+z^2)^n,z^n> in the polynomial ring R=Q[x,y,z]. In her master's thesis [Y], Yao gives a Groebner basis G for q_n and proves that q_n+I_n contain in (q_n:m), where m is the maximal ideal <x,y,z> of R and I_n is the monomial ideal (x^k)(y^k)(z^{2k-1})<x^{2k},y^{2k}><x,y>^{k-1} of R. In this thesis, we prove that (q_n:m) and q_n+I_n are indeed equal. In the process of proving this equality, we give a Groebner basis for the ideals q_n+I_n and find the corner elements of the monomial ideal <LM(q_n)>.

    1. Introduction..............P.1 2. Preliminaries.............P.2 3. A Groebner Basis..........P.9 4. (q4:m)=q4+I4..............P.11 5. Corner Elements...........P.14 6. (q6:m)=q6+I6..............P.26 7. Main Theorem..............P.31 References...................P.70

    1. David Cox, John Little, and Donal O'shea, Ideals, Varieties, and Algorithms, Springer-Verge, Heidelberg, Berlinm, 1992.

    2. L.-M. Li, Some Algorithms of Corner-Elements in Monomial Ideals, master thesis, National Taiwan Normal University, 2007.

    3.Y.-T. Yao, An Application of Groebner Bases : Computing the index of reducibility, master thesis, National Taiwan Normal University, 2009.

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