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.