Let
f
f
be a polynomial of degree
d
d
with integer coefficients,
p
p
any prime,
m
m
any positive integer and
S
(
f
,
p
m
)
S(f,p^m)
the exponential sum
S
(
f
,
p
m
)
=
∑
x
=
1
p
m
e
p
m
(
f
(
x
)
)
S(f,p^m)= \sum _{x=1}^{p^m} e_{p^m}(f(x))
. We establish that if
f
f
is nonconstant when read
(
mod
p
)
\pmod p
, then
|
S
(
f
,
p
m
)
|
≤
4.41
p
m
(
1
−
1
d
)
|S(f,p^m)|\le 4.41 p^{m(1-\frac 1d)}
. Let
t
=
ord
p
(
f
′
)
t=\text {ord}_p(f’)
, let
α
\alpha
be a zero of the congruence
p
−
t
f
′
(
x
)
≡
0
(
mod
p
)
p^{-t}f’(x) \equiv 0 \pmod p
of multiplicity
ν
\nu
and let
S
α
(
f
,
p
m
)
S_\alpha (f,p^m)
be the sum
S
(
f
,
p
m
)
S(f,p^m)
with
x
x
restricted to values congruent to
α
(
mod
p
m
)
\alpha \pmod {p^m}
. We obtain
|
S
α
(
f
,
p
m
)
|
≤
min
{
ν
,
3.06
}
p
t
ν
+
1
p
m
(
1
−
1
ν
+
1
)
|S_\alpha (f,p^m)| \le \min \{\nu ,3.06\} p^{\frac t{\nu +1}}p^{m(1-\frac 1{\nu +1})}
for
p
p
odd,
m
≥
t
+
2
m \ge t+2
and
d
p
(
f
)
≥
1
d_p(f)\ge 1
. If, in addition,
p
≥
(
d
−
1
)
(
2
d
)
/
(
d
−
2
)
p \ge (d-1)^{(2d)/(d-2)}
, then we obtain the sharp upper bound
|
S
α
(
f
,
p
m
)
|
≤
p
m
(
1
−
1
ν
+
1
)
|S_\alpha (f,p^m)| \le p^{m(1-\frac 1{\nu +1})}
.