Abstract
Let n , r n, r and f f be positive integers. Let p p be a prime number and Ï \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 â F q [ x 1 , ⊠, x n ] \{F_i\}_{i=1}^r\subseteq \mathbb F_q[x_1,\dots ,x_n] . Let Z â„ 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 ) = â d α d X d H(X)=\sum _{{\mathbf {d}}}\alpha _{\mathbf {d}}X^{\mathbf {d}} with d = ( d 1 , ⊠, d n ) â Z â„ 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 â F q â \alpha _{\mathbf {d}}\in \mathbb {F}_q^* , we define deg A ⥠( H ) = max d { â i â 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 , Ï ) = â X â V ( F q ) Ï ( 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 , Ï ) â„ | A | â â 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 â { 1 , ⊠, r } i\in \{1,\dots ,r\} . This estimate improves Sperberâs 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 Wanâs results gotten in 1995.
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