Abstract
Let
be a nonsingular quadratic form with integer coefficients,
n
be even. Let
V
=
V
Q
=
V
p
2
denote the set of zeros of
Q
(
x
)
in
Z
p
2
,
p
be an odd prime, and
|
V
|
denote the cardinality of
V
. In this paper, we are interested in giving an upper bound of the number of integer solutions of the congruence
Q
(
x
)
≡
0
(
mod
p
2
)
in small boxes of the type
{
x
∈
Z
p
2
n
|
a
i
⩽
x
i
<
a
i
+
m
i
,
1
⩽
i
⩽
n
}
centered about the origin, where
a
i
,
m
i
∈
Z
, and
0
<
m
i
<
p
2
for
1
⩽
i
⩽
n
.
MSC:
11E04, 11E08, 11E12, 11P21.