Improvements of 𝑝-adic estimates of exponential sums
Authors: Feng Yulu , Hong Shaofang 
Published: February 2022

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold d equals left-parenthesis d 1 comma ellipsis comma d Subscript n Baseline right-parenthesis element-of double-struck upper Z Subscript greater-than-or-equal-to 0 Superscript n Baseline comma upper X Superscript bold d Baseline equals x 1 Superscript d 1 Baseline ellipsis x Subscript n Superscript d Super Subscript n Superscript">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S left-parenthesis upper F comma upper V comma psi right-parenthesis equals sigma-summation Underscript upper X element-of upper V left-parenthesis double-struck upper F Subscript q Baseline right-parenthesis Endscripts psi left-parenthesis upper F left-parenthesis upper X right-parenthesis right-parenthesis">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal o normal r normal d Subscript q Baseline upper S left-parenthesis upper F comma upper V comma psi right-parenthesis greater-than-or-equal-to StartFraction StartAbsoluteValue upper A EndAbsoluteValue minus sigma-summation Underscript i equals 1 Overscript r Endscripts degree Subscript upper A Baseline left-parenthesis upper F Subscript i Baseline right-parenthesis Over max Underscript 1 less-than-or-equal-to i less-than-or-equal-to r Endscripts left-brace right-brace comma of degdegA left-parenthesis right-parenthesis upper F comma of degdegA left-parenthesis right-parenthesis Fi EndFraction">

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.