关于多元多项式模m正交组

ON ORTHOGONAL SYSTEMS OF POLYNOMIALS IN SEVERAL INDETERMINATES OVER Z/mZ

设整数ｍ＞１，ｍ＝是ｍ的标准分解式，１≤ｋ≤ｎ，ｆ＿１，…，ｆ＿ｋ是ｋ个ｎ元整系数多项式。本文证明了：１）ｆ＿１，…，ｆ＿ｋ是模ｍ的正交组当且仅当ｆ＿１，…，ｆ＿ｋ是模的正交组，ｊ＝１，…，ｔ．２）设ｆ＿１，…，ｆ＿ｋ是模ｐ的正交组，且对任一组整数ａ＿１，…，ａ＿ｋ，均有秩（Ｊｍｏｄｐ）＝ｋ，则ｆ＿１，…，ｆ＿ｋ是模ｐ￣ｌ的正交组，ｌ＞１．这里ｐ是一个素数，Ｊ是ｆ＿１…，ｆ＿ｋ的Ｊａｃｏｂｉ矩阵．

Let m>1,1≤k≤n,and In this paper,the authors proved following results : 1)The system off_1 ,…,f_k is orthogonal modulo m if and only ifthe system off_1,…,f_k is orthogonal modulo ,j=1,…,t where are distinctprimes,l_j≥1,j=1,…,t. 2) Let p be a prime and h > 1. If the system of f_1,…,f_k iS orthogonalmodulo p and for any(a_1,…,a_k)∈Z￣(k) the rank of,J mod p=k, then the system of f_1,…,f_k isorthogonal modulo p￣k, where J denotes the Jacobi matrix of f_1,…,f_k