A Curious Identity on Multiple Sums over Fields with Applications

• 作者: Yongchao Xu Shaofang Hong
• 来源期刊: 代数集刊（英文版）
• 年，卷（期）: 2021,（2）
• 页码范围: 295-308
• 页数: 14页
• 语种: 英文
• DOI: 10.1142/S1005386721000237

摘要

Let F be a field,and let e,k be integers such that 1≤e≤|F\{0}| and k≥0.We show that for any subset{a1,…,ae}C F\{0},the curious identity∑(i1,…,ie)∈Ze≥0,i1+…+ie=kai11…aiee=e∑i=1ak+e-1i/e∏i≠j=1((ai-aj))holds with Z≥0 being the set of nonnegative integers.As an application,we prove that for any subset{a1,…,ae}? Fq＼{0}with Fq being the finite field of q elements and e,l being integers such that 2≤e≤q-1 and 0≤l≤e-2,∑(i1,…,ie)∈Ze≥0,i1+…+ie=q-e+lai11…aiee=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{a1,…,ar}? F* we have xq-1-1/r∏i=1(x-ai)=q-1-r∑i=0(∑i1+…+ir=q-1-r-iai11…airr)xi.This implies#{x ∈ F*q | ∑q-1-ri=0(∑i1+…+ir=q-1-r-iai11…airr)xi=0}=q-1-r.