A Curious Identity on Multiple Sums over Fields with Applications

作     者：Yongchao Xu Shaofang Hong Yongchao Xu;Shaofang Hong

作者机构：Mathematical CollegeSichuan UniversityChengdu 610064China 

出 版 物：《Algebra Colloquium》 (代数集刊（英文版）)

年 卷 期：2021年第28卷第2期

页      面：295-308页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：supported partially by the National Science Foundation of China(Grant#11771304) the Fundamental Research Funds for the Central Universities 

主　　题：identity homogeneous polynomial zero polynomial field finite field decomposition generalized principle of cross-classification 

摘      要：Let F be a field,and let e,k be integers such that 1≤e≤|F\{0}|and k≥*** show that for any subset{a1,……,ae}■F\{0},the curious identity∑(i1+……ie)∈Z^(e)≥0,i1+……+ie=k a_(1)^(i1)…a_(e)^(ie)=∑i=1 e a_(i)^(k+e-1)/∏i≠j=1 e(a_(i)-a_(j))holds with Z≥0 being the set of nonnegative *** an application,we prove that for any subset{a_(1)…,a_(e)}■F_(q)\{0}with F_(q)being the finite field of q elements and e,l being integers such that 2≤e≤q-1 and 0≤l≤e-2,∑(i_(1),…,i_(e))∈Z^(e)≥0,i_(1)+…i_(e)=q-e+l a_(1)^(i1)…a_(e)^(ie)=0 Using this identity and providing an extension of the principle of cross-classification that slightly generalizes the one obtained by Hong in 1996,we show that if r is an integer with 1≤r≤q-2,then for any subset{a_(1),…a_(r)}■F_(q)^(*)we have x^(q-1)-1/∏i=1 r(x-a_(i))-∑i=1 q-1-r(∑i_(1)+…+i_(r)=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i).This implies#{x∈Fq*|∑i=0 q-1-4(∑_(i1)+…+ir=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i)=0}=q-1-r.