代数数域上的指数和的估值  

ON EXPONENTIAL SUMS OVER AN ALGEBRAIC NUMBER FIELD

作　　者：方华鹏 

机构地区：[1]武汉大学

出　　处：《数学杂志》1992年第2期121-129,共9页Journal of Mathematics

摘　　要：设 f(x)=a_kx^k+…+a_1x+a_0∈Z[x],a_k≠0,q∈N,(q,a_k,…,a_0)=1,定义指数和:S(f;q)=(?),其中 x 跑遍 mod q 的一个完全剩余系.1940年华罗庚证明了:对于任意实数ε>0均有|S(f;q)|≤c(ε,k)·q^(1-1/k+(?)),其中 c(ε,k)为仅依赖于ε、k的正常数.Let K be an algebraic number field of degree n over the rational field Q,and let δ be the ground ideal(different)of the field.Letf(x)=a_kx^k+…+a_1x+a_o∈K[x],a_k≠0,and let u be the fractional ideal generated by a_k,…,a_1,that is,u=(a_k,…,a_1).Suppose uδ=p/(?),where p and (?) are two relatively prime integral ideals,andS(f(x),(?))= (?),where x runs over a cemplete resiclue system mod(?).If f(x)=a_kx^k+…+a_kx+a_o=a_k(x-ξ_1)~l_1…(x-ξ_m)~lm,where ξ_1,…,ξ_m aredistinct algebraic numbers,then△(f(x))=(?),thenl(f(x)) =max{l_1,l_2,…,l_m}.It is the aim of the paper to prove the following:Theorem:Let ker(?)=(?),where p_i(1≤i≤s)are prime ideals.If(ker(?))~2|(?),|S(f(x),(?))|≤c(ε,k,n)N(?)~1-1/(4l)-+sN(△~4,(?))1/(4l),where l=l(ιf′(x)),△=△(ιf′(x)),ε>0,the constant c(ε,k,n) depends only ε、k、n.τ=μ_iι_iq_i∈Z_k(1≤i≤s)such that the p_i and the content of the polynomialι_iq_i(f′(x)-f′(0))are relatively prime(1≤i≤s).

关 键 词：代数数域 指数和 估值 

分 类 号：O156.2[理学—数学]