On Bernstein-Sato polynomials
On Bernstein-Sato polynomials
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关于 Bernstein-Sato 多项式
DOI：
10.1090/s0002-9939-97-03774-x
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发表时间：
1997
影响因子：
1.3
通讯作者：
G. Lyubeznik
中科院分区：
文献类型：
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作者：
G. Lyubeznik
我们表明，固定的n和d的Bernstein-Sato多项式的所有多项式在最多n个变量的次数最多d的集合是有限的。作为推论，我们证明了存在一个只依赖于n和d的整数t，使得f − t生成R的k-线性微分算子环上的模Rf，其中k是特征为0的任意域，R是k上的n元多项式环，f ∈ R是一个次数至多为d的任意非零多项式。本文中k是一个特征为0的域，Rn（k）= k [X_1，. . . An（k）是相应的Weyl代数，即由Rn（k）的所有元素相乘和n个偏微分d1 = ddX1，.生成的Homk（Rn（k），Rn（k））的子环。. .，dn = ddXn。每个元素Q ∈ An（k）可以唯一地写成有限和Q = ∑ σ i1，.，in，j1，.，jnX i1 1. . . x in n d j11. . . d jn n，其中所有i 1，. . .，in，j1，. . .，jn是非负整数，并且σ i1，.，in，j1，.，jn ∈ k。关于An（k）的基本事实将在下面使用而没有具体的参考;它们都可以在例如[B，pp. 1 - 15]。对于每个多项式f ∈ Rn（k），存在一个函数方程
We show that for fixed n and d the set of Bernstein-Sato polynomials of all the polynomials in at most n variables of degrees at most d is finite. As a corollary, we show that there exists an integer t depending only on n and d such that f−t generates Rf as a module over the ring of the k-linear differential operators of R, where k is an arbitrary field of characteristic 0, R is the ring of polynomials in n variables over k and f ∈ R is an arbitrary non-zero polynomial of degree at most d. Throughout this paper k is a field of characteristic 0, Rn(k) = k[X1, . . . , Xn] is the ring of polynomials in v variables over k and An(k) is the corresponding Weyl algebra, i.e. the subring of Homk(Rn(k), Rn(k)) generated by the multiplications by all the elements of Rn(k) and the n partial differentiations d1 = d dX1 , . . . , dn = d dXn . Every element Q ∈ An(k) may be uniquely written as a finite sum Q = ∑ σi1,...,in,j1,...,jnX i1 1 . . . X in n d j1 1 . . . d jn n where all i1, . . . , in, j1, . . . , jn are non-negative integers and σi1,...,in,j1,...,jn ∈ k. Elementary facts about An(k) will be used below without specific references; they all may be found, for example, in [B, pp. 1–15]. For every polynomial f ∈ Rn(k) there exists a functional equation