Some Estimates For Exponential Sums In Function Fields

In 1997,Chen Jingrun proved the following conclusion:Let n,N?N with n? 3 and N? 2.Let a0,a1,…,an? Z,and write f(x)=anxn+…+a1x+ a0.If gcd(N,an,an-1,…,a1)=1,then|(?)exp(2?(?)/Nf(j))|?CnN1-1/n,where HereC3'=6.1,C4'=5.5:C5'=5,C6'=4.7,C7'=4.4,C8'=4.2,C9'=4.05.In this paper we mainly investigate the analogue of the above result in function fields.Let q ? N+.We denote by Fq the finite field of q elements.Let p be the characteristic of Fq,and let A=Fq[t]denote be the polynomial ring over Fq.For N ? N+,define GN={m ? A:degm<N}.Also,for a ? A,we defineIn 1974,Kubota considered this problem for rnonomials.He gave the following conclusion:Let n?N+with n? 2,a,b?A with b?0.If gcd(a,b)=1 and p(?)n,then|(?)e(adn/b)|?(n-1)(n-1)6|b|1-1/n.In 2012,Zhao Xiaomei proved the following result:Let n N with n? 2.Let b,a0,…,an A with b,an?0:write f(x)=anxn+…+a1x+a0.If p(?)n and gcd(b,an)=1,then for(?)?>0,we have|(?)e(f(d)/b)|?D|b|1-1/2n+?.where,D>0 is a constant depending only on ?,n and q.In this paper we proved the following result:Let n? N with n? 2,a,b ? A with b?0.Let a0,…,an ? A with an?0,write f(x)=an,xn+…+a1x+a0.If n<p and gcd(b.a)=gcd(b,an,an-1.…a1)=1,then|(?)e(a/bf(d))|?Cn|b|1-1/n,whereWhen a=1,an-i=…=a1=a0=0 and p>n? 4,the above conclusion improves the constant in Kubota's result.If one ignores constant factors,it is easy to see that our result improves the conclusion of Zhao Xiaomei in the case when a=1,b(?)Fq and p>n?2.In addition.In this paper we also give integral estimates for a class of exponential sums,and we apply them to obtain the upper bound on the number of solutions of a class of multivariate homogeneous equations.