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=0for
f
<f
. It can easily be
is a root of
g
f
(
X
). Obviously we have
g
f
(
z
0
)
f
since otherwise we had
ρ
f
=1.
Taking into account that the number of distinct roots of
Ψ
f
1
,...,f
2
ν
(
X
)isless
than 2
νh
anditcannotbean
s
th-power, Lemma 2 applies. We obtain that the
total contribution from such terms is
O
(
L
2
ν
hp
1
/
2
). Hence
=0for
f
≤
seen with (7) that
h
f
(
z
0
)
2
ν
=
O
T
2
ν−
1
L
ν
p
+
L
2
ν
hp
1
/
2
.
L
2
ν
|
S
χ
|
So this leads us to the bound
2
ν
=
O
T
2
ν−
1
L
−ν
p
+
hp
1
/
2
.
|
S
χ
|
Recalling that
L
hT/t
,wederive
2
ν
=
O
T
2
ν−
1
t
ν
T
−ν
h
−ν
p
+
hp
1
/
2
.
|
S
χ
|
Substituting the selected value of
h
, which balances both terms in the above
estimate, we finish the proof.
Remark.
As for the bound for Dickson polynomials we mention the following
simplifications. Assuming that
T
=
t
1+
o
(1)
, the bound of Theorem 1 takes the
form
S
χ
=
O
T
1
−
1
/
2
ν
+
o
(1)
p
(
ν
+2)
/
4
ν
(
ν
+1)
.
Therefore for any
δ>
0, choosing a suciently large
ν
we obtain a nontrivial
bound provided
T
p
1
/
2+
δ
. On the other hand, if
t
T
=
p
1+
o
(1)
,thentaking
≥
≥
ν
=1weobtain
S
χ
=
O
p
7
/
8+
o
(1)
.
Acknowledgment
Parts of this paper were written during a pleasant stay of D.G. to Linz. He
wishes to thank the Radon Institute for its hospitality. A.W. was supported in
part by the Austrian Science Fund (FWF) Grant P19004-N18.
References
1. Friedlander, J.B., Hansen, J., Shparlinski, I.E.: On character sums with exponential
functions. Mathematika 47, 75-85 (2000)
2. Gomez, D., Winterhof, A.: Character sums for sequences of iterations of Dick-
son polynomials. In: Mullen, G., et al. (eds.) Finite Fields and Applications Fq8,
Contemporary Mathematics (to appear)
3. Korobov, N.M.: The distribution of non-residues and of primitive roots in recur-
rence series. Dokl. Akad. Nauk SSSR 88, 603-606 (1953) (in Russian)
