Improvements of p p -adic estimates of exponential sums

Let n,rn, r and ff be positive integers. Let pp be a prime number and ψ\psi be an arbitrary fixed nontrivial additive character of the finite field Fq\mathbb F_q with q=pfq=p^f elements. Let FF be a polynomial in Fq[x1,…,xn]\mathbb F_q[x_1,\dots ,x_n] and VV be the affine algebraic variety defined over Fq\mathbb {F}_q by the simultaneous vanishing of the polynomials {Fi}i=1r⊆Fq[x1,…,xn]\{F_i\}_{i=1}^r\subseteq \mathbb F_q[x_1,\dots ,x_n]. Let Z≥0\mathbb {Z}_{\ge 0} stand for the set of all nonnegative integers and AA be an arbitrary nonempty subset of {1,…,n}\{1,\dots ,n\}. For a polynomial H(X)=∑dαdXdH(X)=\sum _{{\mathbf {d}}}\alpha _{\mathbf {d}}X^{\mathbf {d}} with d=(d1,…,dn)∈Z≥0n,Xd=x1d1…xndn{\mathbf {d}}=(d_1,\dots ,d_n)\in \mathbb {Z}_{\ge 0}^n, X^{\mathbf {d}}=x_1^{d_1}\dots x_n^{d_n} and αd∈Fq∗\alpha _{\mathbf {d}}\in \mathbb {F}_q^*, we define degA⁡(H)=maxd{∑i∈Adi}\deg _A(H)=\max _{{\mathbf {d}}}\{\sum _{i\in A}d_i\} to be the AA-degree of HH. In this paper, for the exponential sum S(F,V,ψ)=∑X∈V(Fq)ψ(F(X))S(F,V,\psi )=\sum _{X\in V(\mathbb {F}_q)}\psi (F(X)) with V(Fq)V(\mathbb {F}_q) being the set of the Fq\mathbb {F}_q-rational points of VV, we show that ordqS(F,V,ψ)≥|A|−∑i=1rdegA⁡(Fi)max1≤i≤r{degA⁡(F),degA⁡(Fi)}\begin{equation*} \mathrm {ord}_q S(F,V,\psi )\ge \frac {|A|-\sum _{i=1}^r\deg _A(F_i)} {\max _{1\le i\le r}\{\deg _A(F),\deg _A(F_i)\}} \end{equation*} if degA⁡(F)>0\deg _A(F)>0 or degA⁡(Fi)>0\deg _A(F_i)>0 for some i∈{1,…,r}i\in \{1,\dots ,r\}. This estimate improves Sperber’s theorem obtained in 1986. This also leads to an improvement of the pp-adic valuation of the number N(V)N(V) of Fq\mathbb {F}_q-rational points on the variety VV which strengthens the Ax-Katz theorem. Moreover, we use the AA-degree and pp-weight AA-degree to establish pp-adic estimates on multiplicative character sums and twisted exponential sums which improve Wan’s results gotten in 1995.