arXiv:1703.07066v2 [math.NT] 27 Mar 2017
BOUNDS ON EXPONENTIAL SUMS WITH
QUADRINOMIALS
SIMON MACOURT
Abstract. We improve an existing result on exponential quadrilinear sums in the case of sums over multiplicative subgroups of
a finite field and use it to give a new bound on exponential sums
with quadrinomials.
1. Introduction
1.1. Set Up. For a prime p, we use Fp to denote the finite field of p
elements.
For a t-sparse polynomial
ΨpXq “
t
ÿ
ai X ki
i“1
with some pairwise distinct non-zero integer exponents k1 , . . . , kt and
coefficients a1 , . . . , at P F˚p , and a multiplicative character χ of F˚p we
define the sums
ÿ
Sχ pΨq “
χpxq ep pΨpxqq,
xPF˚
p
where ep puq “ expp2πiu{pq and χ is an arbitrary multiplicative character of F˚p . The challenge for such sums is to provide a bound that is
stronger than the Weil bound
Sχ pΨq ď maxtk1 , . . . , kt up1{2 ,
see [18, Appendix 5, Example 12], by taking advantage of the arithmetic
structure of the exponents. The case of exponential sums of monomials has seen much study with Shparlinski [16] providing the first such
bound. Further improvements have been made by various other authors, see [1, 3, 11, 12, 15, 17]. We also mention that Cochrane, Coffelt
and Pinner, as well as others, have given several bounds on exponential
sums with sparse polynomials, see [4–9] and references therein, some
of which we outline in Section 1.2.
2010 Mathematics Subject Classification. 11L07, 11T23.
Key words and phrases. exponential sum, sparse polynomial, quadrinomial.
1
2
S. MACOURT
Here we provide some new bounds on quadrinomial exponential sums
using the techniques in [13]. We thus define
ΨpXq “ aX k ` bX ℓ ` cX m ` dX n .
(1.1)
We mention that all our results extend naturally to more general sums
with polynomials of the shape
ΨpXq “ aX k ` f pX ℓ q ` gpX mq ` hpX n q
for polynomials f, g, h P Fp rXs.
The notation A ! B is equivalent to |A| ď c|B| for some constant
c.
1.2. Previous Results. We compare our result for quadrinomials (1.1)
to those of Cochrane, Coffelt and Pinner [4, Theorem 1.1]
ˆ
˙1{9
kℓmn
Sχ pΨq !
p8{9
maxpk, ℓ, m, nq
which is non-trivial for
kℓmn
ă p,
maxpk, ℓ, m, nq
and of Cochrane and Pinner [6, Theorem 1.1]
Sχ pΨq ! pkℓmnq1{16 p7{8
which is non-trivial for kℓmn ă p2 . Our new result in Theorem 1.1 is
independent of the size of the exponents but instead depends on various
greatest common divisors.
1.3. Main Result. Our main result is the following theorem.
Theorem 1.1. Let ΨpXq be a quadrinomial of the form (1.1) with
a, b, c, d P F˚p . Define
α “ gcdpk, p´1q, β “ gcdpℓ, p´1q, γ “ gcdpm, p´1q, δ “ gcdpn, p´1q
and
α
β
,
g“
,
gcdpα, δq
gcdpβ, δq
Suppose f ě g ě h, then p{δ ě f and
f“
Sχ pΨq !pg ´1{8
$ 15{16 1{32
p
δ ,
’
’
& 31{32 1{32 ´1{16`op1q
p
δ g
,
`
1{32
´1{16`op1q
pδ
pf
gq
,
’
’
% 31{32`op1q 3{32
p
δ pf gq´1{16,
h“
if
if
if
if
γ
.
gcdpγ, δq
g ě p1{2 log p,
f ě p1{2 log p ą g,
p{δ ě p1{2 log p ą f ,
p{δ ă p1{2 log p.
BOUNDS ON EXPONENTIAL SUMS WITH QUADRINOMIALS
3
We mention that our result is independent of the size of our powers
k, l, m, n and is strongest when δ is small and f, g, h are large. As
mentioned in the previous section, previous results become trivial for
quadrinomials of large degree. It is easy to see that our bound is nontrivial and improves previous results for a wide range of exponents
k, ℓ, m and n.
2. Preliminaries
We recall the following classical bound of bilinear sums, see, for
example, [2, Equation 1.4] or [10, Lemma 4.1].
Lemma 2.1. For any sets X , Y Ď Fp and any α “ pαx qxPX , β “
pβy qyPY , with
ÿ
ÿ
|αx |2 “ A
and
|βy |2 “ B,
xPX
we have
yPY
ˇ
ˇ
ˇ a
ˇÿ ÿ
ˇ
ˇ
αx βy ep pxyqˇ ď pAB.
ˇ
ˇ
ˇxPX yPY
We define Dˆ pUq to be the number of solutions of
pu1 ´ v1 qpu2 ´ v2 q “ pu3 ´ v3 qpu4 ´ v4 q,
ui , vi P U, i “ 1, 2, 3, 4.
ˆ
We also define the multiplicative energy E pU, Vq to be the number of
solutions of
u1 v1 “ u2 v2
ui P U, vi P V, i “ 1, 2.
ˆ
When U “ V , we write E pU, Uq “ E ˆ pUq.
We need the following result from [13, Corollary 3.3].
Lemma 2.2. For a multiplicative subgroup G Ă F˚p , we have
"
|G|8 p´1 ,
if |G| ě p1{2 log p,
Dˆ pGq !
|G|6 log |G|, if |G| ă p1{2 log p.
We also use [13, Corollary 4.1].
Lemma 2.3. Let G be a multiplicative subgroup of F˚p . Then for any
λ P F˚p , we have
$ 1{2 3{2
& p |G| , if |G| ě p2{3 ,
4
|G|
ˆ
!
E pG ` λq ´
|G|3 p´1{2 ,
if p2{3 ą |G| ě p1{2 log p,
%
p
2
|G| log |G|, if |G| ă p1{2 log p.
We immediately obtain the following result by observing the dominant term from Lemma 2.3.
4
S. MACOURT
Corollary 2.4. Let G be a multiplicative subgroup of F˚p . Then for any
λ P F˚p , we have
" 4
if |G| ě p1{2 log p,
|G |{p,
ˆ
E pG ` λq !
2
|G| log |G|, if |G| ă p1{2 log p.
We define NpF , G, Hq to be the number of triples of solutions to
f1 pg1 ´ g2 q “ f2 ph1 ´ h2 q where fi P F , gi P G, hi P H for i “ 1, 2.
Using Corollary 2.4 we obtain the following result.
Lemma 2.5. Let F , G, H be multiplicative subgroups of F˚p with cardinalities F, G, H respectively with G ě H . Additionally, let M “
maxpF, Gq. Then
$ 2 2 ´1{2
,
if H ě p1{2 log p,
2 & G H p
F
NpF , G, Hq ! 1{2
G2 H 3{2`op1q p´1{4 , if G ě p1{2 log p ą H,
M %
pGHq3{2`op1q ,
if G ă p1{2 log p.
Proof. By multiplying both sides of f1 pg1 ´ g2 q “ f2 ph1 ´ h2 q by the
inverses f2´1 and h´1
2 and taking a factor of g2 from the left hand side,
and defining S “ tf gh : f P F , g P G, h P Gu we have
F 2 GH ÿ
|tλpg ´ 1q “ h ´ 1 : g P G, h P Hu|.
NpF , G, Hq “
|S| λPS
By the Cauchy inequality,
ˇ"
*ˇ
ˇ
F 4 G2 H 2 ˇˇ h1 ´ 1
h2 ´ 1
2
NpF , G, Hq ď
“
: hi P H, gi P G, i “ 1, 2 ˇˇ
ˇ
|S|
g1 ´ 1
g2 ´ 1
4 2 2
F G H
pEˆ pG ´ 1qEˆ pH ´ 1qq1{2 .
“
|S|
By Corollary 2.4,
$ 2 2
if H ě p1{2 log p,
4 2 2 & G H {p,
F
G
H
NpF , G, Hq2 !
G2 H 1`op1q p´1{2 , if G ě p1{2 log p ą H,
|S| %
pGHq1`op1q ,
if G ă p1{2 log p.
Since |S| ě M we complete our proof.
\
[
Applying Lemma 2.2 and Lemma 2.5 in the proof of [14, Theorem
1.4], we obtain the following result on quadrilinear sums over subgroups.
Lemma 2.6. For any multiplicative subgroups W, X , Y, Z Ď F˚p of cardinalities W, X, Y, Z , respectively, with W ě X ě Y ě Z and weights
ϑ “ pϑw,x,y q, ρ “ pρw,x,z q, σ “ pσw,y,z q and τ “ pτx,y,z q with
max
pw,x,yqPWˆX ˆY
|ϑw,x,y | ď 1,
max
pw,x,yqPWˆX ˆZ
|ρw,x,z | ď 1,
BOUNDS ON EXPONENTIAL SUMS WITH QUADRINOMIALS
max
pw,x,yqPWˆYˆZ
|σw,y,z | ď 1,
max
pw,x,yqPX ˆYˆZ
5
|τx,y,z | ď 1,
for the sums
T “
ÿ ÿÿÿ
ϑw,x,y ρw,x,z σw,y,z τx,y,z ep pawxyzq
wPW xPX yPY zPZ
we have
|T | ! W XZY 7{8
$ 31{32
W
XY Zp´1{32 ,
’
’
& 31{32
W
XY 15{16`op1q Z,
`
W 31{32 pXY q15{16`op1q Zp1{32 ,
’
’
% 29{32`op1q
W
pXY q15{16 Zp1{16 ,
if
if
if
if
Y ě p1{2 log p,
X ě p1{2 log p ą Y ,
W ě p1{2 log p ą X,
W ă p1{2 log p.
uniformly over a P F˚p .
Proof. We see from [14, p. 24] that
ÿ ÿ
JpµqIpλqηµ ep pλµq ` pW XZq8 Y 7 ,
|T |8 ! pW XY q6 Z 7
µPF˚
p λPFp
where ηµ , µ P F˚p is a complex number with |ηµ | “ 1, Jpµq is the
number of quadruples px1 , x2 , y1, y2 q P X 2 ˆY 2 such that px1 ´x2 qpy1 ´
y2 q “ µ P F˚p and Ipλq is the number of triples pw1 , w2 , zq P W 2 ˆ Z
such that zpw1 ´ w2 q “ λ P Fp . We estimate Jpµq as in [14, Equation
3.10] but using our bound from Lemma 2.2 to obtain
$ 4 4
if Y ě p1{2 log p,
& X Y {p,
ÿ
(2.1)
Jpµq2 !
X 4 Y 3`op1q p´1{2 , if X ě p1{2 log p ą Y ,
%
µPF˚
pXY q3`op1q ,
if X ă p1{2 log p.
p
Now
ÿ
Ipλq2
λPFp
“ |tz1 pw1 ´ w2 q “ z2 pw3 ´ w4 q : w1 , w2 P W, zi P Z, i “ 1, 2, 3, 4u|
“ NpZ, W, W q.
Therefore, by Lemma 2.5,
" 2 7{2 ´1{2
ÿ
Z W p
, if W ě p1{2 log p,
2
(2.2)
Ipλq !
2
5{2`op1q
Z W
, if W ă p1{2 log p.
λPFp
6
S. MACOURT
Applying the classical bound on bilinear exponential sums from Lemma
2.1 together with (2.1) and (2.2), we get
|T |8 !pW XZq8Y 7
$ 31{4 8 8 8 ´1{4
W
X Y Z p
,
’
’
& 31{4 8 15{2`op1q 8
W
X Y
Z ,
`
W 31{4 XpY Zq15{2`op1q Z 8 p1{4 ,
’
’
% 29{4`op1q
W
pXY q15{2 Z 8 p1{2 ,
if
if
if
if
Y ě p1{2 log p,
X ě p1{2 log p ą Y ,
W ě p1{2 log p ą X,
W ă p1{2 log p.
|T | !W XZY 7{8
$ 31{32
W
XY Zp´1{32 ,
’
’
& 31{32
W
XY 15{16`op1q Z,
`
31{32
W
pXY q15{16`op1q Zp1{32 ,
’
’
% 29{32`op1q
W
pXY q15{16 Zp1{16 ,
if
if
if
if
Y ě p1{2 log p,
X ě p1{2 log p ą Y ,
W ě p1{2 log p ą X,
W ă p1{2 log p.
Hence,
This completes the proof.
\
[
We compare our bound for subgroups from Lemma 2.6 with that for
arbitrary sets coming from [14, Theorem 1.4]
ˇ
ˇ
ˇ
ˇÿ ÿ ÿÿ
ˇ
ˇ
ϑw,x,y ρw,x,z σw,y,z τx,y,z ep pawxyzqˇ
ˇ
ˇ
ˇwPW xPX yPY zPZ
(2.3)
! p1{16 W 15{16 pXY q61{64 Z 31{32 .
For example, if W “ X “ Y “ Z “ p1{2`op1q then the bounds become
p125{64`op1q and p63{32`op1q respectively.
3. Proof of Theorem 1.1
Let Gα , Gβ , Gγ be the subgroups of F˚p formed by the elements of
orders α, β and γ respectively. Then,
1 ÿ ÿ ÿ ÿ
χpwxyzq ep pΨpwxyzqq
Sχ pΨq “
αβγ xPG yPG zPG
˚
α
“
1
αβγ
β
γ
wPFp
ÿ ÿ ÿ ÿ
xPGα yPGβ zPGγ wPF˚
p
χpwxyzq ep paw k y k z k ` bw ℓ xℓ z ℓ ` cw m xm y m ` dw n xn y n z n q
1 ÿ ÿ ÿ ÿ
ϑw,x,y ρw,x,z σw,y,z ep pdw n xn y n z n q
“
αβγ xPG yPG zPG
˚
α
β
γ
wPFp
BOUNDS ON EXPONENTIAL SUMS WITH QUADRINOMIALS
7
where ϑw,x,y “ χpwxyq ep pcw m xm y m q, ρw,x,z “ χpzq ep pbw ℓ xℓ z ℓ q and
σw,y,z “ ep paw k y k z k q. Now the image W “ tw n : w P F˚p u of non-zero
nth powers contains pp ´ 1q{δ elements, each appearing with multiplicity δ . Similarly, we can see that the images X “ txn : x P Gα u, Y “
ty n : y P Gβ u and Z “ tz n : z P Gγ u contain f, g and h elements
with multiplicity gcdpα, δq, gcdpβ, δq and gcdpγ, δq respectively. We
apply Lemma 2.6, recalling our assumption that f ě g and noticing
f δ “ lcmpα, δq ă p ´ 1, hence f ď p{δ , which gives us
Sχ pΨq !
δ gcdpα, δq gcdpβ, δq gcdpγ, δq
pp{δqf g 7{8h
αβγ
δ gcdpα, δq gcdpβ, δq gcdpγ, δq
`
αβγ
$
pp{δq31{32 f ghp´1{32 ,
if g ě p1{2 log p,
’
’
&
31{32
15{16`op1q
pp{δq
fg
h,
if f ě p1{2 log p ą g,
ˆ
pp{δq31{32 pf gq15{16`op1q hp1{32 , if p{δ ě p1{2 log p ą f ,
’
’
%
pp{δq29{32`op1q pf gq15{16 hp1{16 , if p{δ ă p1{2 log p.
“pg ´1{8
$ 15{16 1{32
p
δ ,
’
’
& 31{32 1{32 ´1{16`op1q
p
δ g
,
`
1{32
´1{16`op1q
pδ pf gq
,
’
’
% 31{32`op1q 3{32
p
δ pf gq´1{16 ,
if
if
if
if
g ě p1{2 log p,
f ě p1{2 log p ą g,
p{δ ě p1{2 log p ą f ,
p{δ ă p1{2 log p.
This concludes the proof.
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8
S. MACOURT
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Department of Pure Mathematics, University of New South Wales,
Sydney, NSW 2052, Australia
E-mail address: [email protected]