一般子式理想之Grbner基底
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2005
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Abstract
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Grbner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Grbner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Grbner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 Grbner 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 Grbner 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.
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 Grbner 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 Grbner 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.
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Grbner基底, 子式, Grbner bases, minor