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研究生： 賴勇仁
Yung-Ren Lai
論文名稱： 一般子式理想之Grbner基底
Grbner bases of ideals of generic minors
指導教授： 洪有情
Hung, Yu-Ching
學位類別： 碩士
Master
系所名稱： 數學系
Department of Mathematics
論文出版年： 2005
畢業學年度： 93
語文別： 英文
論文頁數： 51
中文關鍵詞： Grbner基底 、子式
英文關鍵詞： Grbner bases, minor
論文種類： 學術論文
相關次數： 點閱：108  下載：6 
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    • 設X是一個各個位置為變數x_ij的矩陣，R=K[X]是一個係數佈於一個體的多項式環。在1989年和1990年，Sturmfels，Caniglia和Guccione各自證明了X的所有相同次數的子式對於某個lexicographic單項式次序會是一組Grbner基底；在1992年，Herzog和Trung進一步提供了一種取不同次數的子式也會是Grbner基底的方法。在這篇論文中，我們又提供了一種取不同次數的子式也會是Grbner基底的方法。

    Let K be a field and R=K[X] be the polynomial algebra generated by the entries of a generic m×n matrix X=(x_ij) over K. Let p be a positive integer. Let G_p be the set of all p-minors of X and I be the ideal generated by G_p. Sturmfels and Caniglia et al. had proved that G_p is a Grbner basis for I with respect to some lexicographical term order of R. Later in 1992, Herzog and Trung improved their result. Also, in 1994 Conca obtained a similar result for a symmetric matrix. In this paper, we get some results similar to their results as follows.
    Theorem：Let X=(x_ij) be a generic m×n matrix over a field K, and let R=K[X]. Let m≧a_1≧…≧a_r , b_1≦…≦b_r≦n be nonnegative integers, and η_1,…,η_(r+1) be positive integers. Let D_t(X) be the part of the matrix X consisting of the last a_t rows and the first b_t columns. Let G_t(X) be the set of all (η_t)-minors of D_t(X), t=1,…,r and set D_(r+1)(X) be the set of all (η_(r+1))-minors of X. Let I be the ideal of R generated by the G(X)=∪G_t(X); then G(X) is a Grbner basis for I with respect to the lexicographic term order induced from the variable order
    x_11> x_12>…> x_1n> x_21>… > x_m1>… > x_mn.
    We also prove that if X=(x_ij) in the above theorem is an n×n symmetric matrix, then the theorem also holds.

    Introduction..............................................1 Preliminary...............................................4 Grbner basis for minors of a fixed size of a generic matrix....................................................9 Some useful propositions.................................17 Grbner basis for ladder determinantal ideals............20 Grbner basis for determinantal ideals...................27 Reference................................................51

    Willian W. Adams and Philippe Loustaunau, An Introduction
    to Grbner Bases. American Mathematical Society, 1994.

    L. Caniglia, J. Stein and J.J. Guccione, Ideals of
    generic minors, Comm. in Algebra 18(8) (1990), 2633-2640.

    H.-C. Chao, Grbner bases of Pfaffians, master
    thesis, 2002.

    A. Conca, Grbner bases of ideals of minors of a
    symmetric matrix, J. Algebra 166 no.2, (1994), 406-421.

    J. Herzog and N.V. Trung, Grbner bases and multiplicity
    of determinantal and pfaffian ideals, Adv. in Math. 96 (1992),
    1-37.

    B. Sturmfels, Grbner bases and invariant theory, Adv.
    in Math. 76 (1989), 245-259.

    H.-J. Wang, Grbner bases of ideals generated by
    minors, preprint.

    L.-H. Wu, Grbner bases and pfaffians of
    skew-symmetric matrices, master thesis, 2004.

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