The Coefficients of the Hilbert Polynomial and the Reduction Number of an Ideal
The Coefficients of the Hilbert Polynomial and the Reduction Number of an Ideal
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DOI：
10.1112/jlms/s2-40.1.1
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发表时间：
1989-08
影响因子：
1.2
通讯作者：
T. Marley
中科院分区：
文献类型：
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作者：
T. Marley
设（R，m）是具有极大理想m的n维Cohen-Macaulay（简称CM）局部环.如果f是m-准素理想，我们设f（n）= X（R/P）（这里A（）表示长度），约定P= R，n^ 0。我们称h（n）为f的希尔伯特函数。塞缪尔证明了存在一个形式为fn+ d-1\ fn+ d-2 e\d）-n·（d-1）的多项式Pi（n），使得Pi（n）= Hi（n），其中n> 0。我们称Pi（ri）为f的希尔伯特多项式，并且整数e0，.，艾德（有时写作eo（I），.，艾德（I））称为f的希尔伯特系数。第一个系数e0被称为f的多重性，已经被广泛研究。但对其他系数的重要性知之甚少。参见[1，4，5，13]。
Let (R, m) be a^/-dimensional Cohen-Macaulay (CM for short) local ring with maximal ideal m. If/is an m-primary ideal, we set//,(«)= X (R/P)(here A () denotes length) with the convention that P= R for n^ 0. We call H,(n) the Hilbert function of/. Samuel proved that there exists a polynomial P,(n) of the form fn+ d-1\ fn+ d-2 e\d)-«•(d-\such that P,{n)= H,(n) for n> 0. We call P,(ri) the Hilbert polynomial of/and the integers eo,..., ed(sometimes written eo (I),..., ed (I)) are called the Hilbert coefficients of/. The first coefficient, e0, is called the multiplicity of/and has been studied extensively. But not much is known about the significance of the other coefficients. See [1, 4, 5, 13], however.