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Let n, r and f be positive integers. Let p be a prime number and \psi be an arbitrary fixed nontrivial additive character of the finite field \mathbb F_q with q=p^f elements. Let F be a polynomial in \mathbb F_q[x_1,\dots,x_n] and V be the affine algebraic variety defined over \mathbb {F}_q by the simultaneous vanishing of the polynomials \{F_i\}_{i=1}^r\subseteq \mathbb F_q[x_1,\dots,x_n]. Let \mathbb {Z}_{\ge 0} stand for the set of all nonnegative integers and A be an arbitrary nonempty subset of \{1,\dots,n\}. For a polynomial H(X)=\sum _{{\mathbf {d}}}\alpha _{\mathbf {d}}X^{\mathbf {d}} with {\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 \alpha _{\mathbf {d}}\in \mathbb {F}_q^*, we define \deg _A(H)=\max _{{\mathbf {d}}}\{\sum _{i\in A}d_i\} to be the A-degree of H. In this paper, for the exponential sum S(F,V,\psi)=\sum _{X\in V(\mathbb {F}_q)}\psi (F(X)) with V(\mathbb {F}_q) being the set of the \mathbb {F}_q-rational points of V, we show that \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 or \deg _A(F_i)>0 for some i\in \{1,\dots,r\}. This estimate improves Sperber's theorem obtained in 1986. This also leads to an improvement of the p-adic valuation of the number N(V) of \mathbb {F}_q-rational points on the variety V which strengthens the Ax-Katz theorem. Moreover, we use the A-degree and p-weight A-degree to establish p-adic estimates on multiplicative character sums and twisted exponential sums which improve Wan's results gotten in 1995. [ABSTRACT FROM AUTHOR]

For any prime number p, and integer k ≥ 1, let Fpk be the finite field of pk elements. A famous problem in the theory of polynomials over finite fields is the characterization of all nonconstant polynomials F ∈ Fpk[x] for which the value set {F(α): α ∈ Fpk} has the minimum possible size ⌊(pk−1)/deg F⌋ + 1. For k ≤ 2, the problem was solved in the early 1960s by Carlitz, Lewis, Mills, and Straus. This paper solves the problem for k = 3. [ABSTRACT FROM AUTHOR]