Improvements of 𝑝-adic estimates of exponential sums

Abstract

Let n , r n, r and f f be positive integers. Let p p be a prime number and ψ \psi be an arbitrary fixed nontrivial additive character of the finite field F q \mathbb F_q with q = p f q=p^f elements. Let F F be a polynomial in F q [ x 1 , … , x n ] \mathbb F_q[x_1,\dots ,x_n] and V V be the affine algebraic variety defined over F q \mathbb {F}_q by the simultaneous vanishing of the polynomials { F i } i = 1 r ⊆ F q [ x 1 , … , x n ] \{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 A A be an arbitrary nonempty subset of { 1 , … , n } \{1,\dots ,n\} . For a polynomial H ( X ) = ∑ d α d X d H(X)=\sum _{{\mathbf {d}}}\alpha _{\mathbf {d}}X^{\mathbf {d}} with d = ( d 1 , … , d n ) ∈ Z ≥ 0 n , X d = x 1 d 1 … x n d n {\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 α d ∈ F q ∗ \alpha _{\mathbf {d}}\in \mathbb {F}_q^* , we define deg A ⁡ ( H ) = max d { ∑ i ∈ A d i } \deg _A(H)=\max _{{\mathbf {d}}}\{\sum _{i\in A}d_i\} to be the A A -degree of H H . In this paper, for the exponential sum S ( F , V , ψ ) = ∑ X ∈ V ( F q ) ψ ( F ( X ) ) S(F,V,\psi )=\sum _{X\in V(\mathbb {F}_q)}\psi (F(X)) with V ( F q ) V(\mathbb {F}_q) being the set of the F q \mathbb {F}_q -rational points of V V , we show that o r d q S ( F , V , ψ ) ≥ | A | − ∑ i = 1 r deg A ⁡ ( F i ) max 1 ≤ i ≤ r { deg A ⁡ ( F ) , deg A ⁡ ( F i ) } \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 deg A ⁡ ( F ) > 0 \deg _A(F)>0 or deg A ⁡ ( F i ) > 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 p p -adic valuation of the number N ( V ) N(V) of F q \mathbb {F}_q -rational points on the variety V V which strengthens the Ax-Katz theorem. Moreover, we use the A A -degree and p p -weight A A -degree to establish p p -adic estimates on multiplicative character sums and twisted exponential sums which improve Wan’s results gotten in 1995.