Polynomials arising in factoring generalized Vandermonde determinants: An algorithm for computing their coefficients
Polynomials arising in factoring generalized Vandermonde determinants: An algorithm for computing their coefficients
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DOI：
10.1016/s0895-7177(01)00060-7
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发表时间：
2001-08-01
影响因子：
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通讯作者：
De Marchi, S
中科院分区：
文献类型：
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作者：
De Marchi, S
我们考虑V-s;mu（x（1），. x（s））= /x（i）（muk）/，1小于或等于i，k小于或等于s，其中x（i）是属于真实的线的区间[a，B]的不同点，索引s代表阶，序列mu由0小于或等于mu（1）< mu（2）<.的有序整数组成。< mu（s）。这些行列式可以分解为经典的范德蒙行列式和所涉及的点的齐次对称函数的乘积，即舒尔函数。另一方面，我们证明了当x = x（s）时，在所得到的多项式中，取决于变量x，Schur函数可以被分解为两因子多项式：第一个是常数Pi（s-1）（i=1）x（i）（mu 1）乘以（monic）多项式Pi（s-）（i=1）1（x-x（i））：第二类是M = m（s-1）- s +1次多项式P-M（x）.我们的主要结果是一元多项式PM（s）的系数的计算.基于Schur函数的Jacobi-Trudi恒等式，提出了一种计算P-M系数的算法. (C)2001爱思唯尔科技有限公司版权所有。
We consider generalized Vandermonde determinants of the formV-s;mu(x(1),...x(s)) = /x(i)(muk)/, 1 less than or equal to i, k less than or equal to s,where the x(i) are distinct points belonging to an interval [a, b] of the real line, the index s stands for the order, the sequence mu consists of ordered integers 0 less than or equal to mu (1) < mu (2) < ... < mu (s). These determinants can be factored as a product of the classical Vandermonde determinant and a homogeneous symmetric function of the points involved, that is, a Schur function. On the other hand, we show that when x = x(s), in the resulting polynomial, depending on the variable x, the Schur function can be factored as a two-factors polynomial: the first is the constant Pi (s-1)(i=1) x(i)(mu1) times the (monic) polynomial Pi (s-)(i=1)1 (x -x(i)): while the second is a polynomial P-M(x) of degree M = m(s-1) - s + 1.Our main result is then the computation of the coefficients of the monic polynomial PM(s). We present an algorithm for the computation of the coefficients of P-M based on the Jacobi-Trudi identity for Schur functions. (C) 2001 Elsevier Science Ltd. All rights reserved.