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 我們令K是一個體與X是一個佈於K的m ×n矩陣。 再令R=K[X]與I是一個由X中(r+1) ×(r+1)子行列式所生成的理想。 自從1950年,有一些方法證明了上面的公式,然而,這些方法卻都不容易瞭解。 本篇論文中我們使用了單純複形去簡化我們的難度。 我們就可以很迅速地計算R的重數。
 Let K be a field and X be a generic m ×n matrix over K. Let R=K[X] and I be the ideal generated by the (r+1) ×(r+1) minors of X. There are several ways to prove the above formula since 1950, however, they are not easy to understand. In this paper, we use the simplicial complexes to simplify our difficulty. We are able to quickly calculate the multiplicity of R.
 1. Introduction 2 2. Basic theory of the Stanley-Reisner rings 8 2.1 Simplicial complexes -----------------------------------8 2.2 Stanley-Reisner rings and f-vecters -------------------10 2.3 Hilbert series of the Stanley-Reisner rings -----------11 3. The multiplicity of determinantal ideals 17 3.1 The multiplicity of ideals generated by 2 ×2 minors --17 3.2 The multiplicity of ideals generated by 3 ×3 minors --23 3.3 The multiplicity of ideals generated by maximal minors 28 4. Bibliography 32
 [1] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1993.[2] J. Herzog and N.V. Trung, Grobner bases and multiplicity of determinantal and pfaffian ideals, Adv. in Math, 96(1992), 1-37[3] H. Matsumura, Commutative ring theory, Cambridge University Press, 1986.[4] G.Z. Giambelli, Risoluzione del problema generale numerativo per gli spazi plurisecanti di una curva algebraica, Mem. Acad. Sci. Torino (2) 59, 433-508, 1909.[5] T.-Y Huang, Hilbert Functions of Projective Varieties Defined by Maximal Minors of m by m+1 Matrices, Chia-Yi, 2000[6] C.-S Yu, Hilbert Functions of Projective Varieties Defined by Maximal Minors of 2 by n Matrices, Chia-Yi, 2000[7] S.-F Yang, Hilbert Polynomials of Certain Pfaffian Ideals, Chia-Yi, 2001[8] D.-Y Yan, Hilbert functions of projective varieties defined by 2 by 2 minors of m ×n matrices, Chia-Yi, 2001
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 1 對稱加權矩陣行列式理想的重數

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