On three-variable expanders over finite valuation rings
Author:
Ham Le Quang1, Van The Nguyen1, Tran Phuc D.2, Vinh Le Anh3
Affiliation:
1. VNU University of Science , Vietnam National University , Hanoi , Vietnam 2. Department of Mathematics & Sciences , American University in Bulgaria , Blagoevgrad , Bulgaria 3. Vietnam National University ; and Vietnam Institute of Educational Sciences , Hanoi , Vietnam
Abstract
Abstract
Let
ℛ
{\mathcal{R}}
be a finite valuation ring of order
q
r
{q^{r}}
.
In this paper, we prove that for any quadratic polynomial
f
(
x
,
y
,
z
)
∈
ℛ
[
x
,
y
,
z
]
{f(x,y,z)\in\mathcal{R}[x,y,z]}
that is of the form
a
x
y
+
R
(
x
)
+
S
(
y
)
+
T
(
z
)
{axy+R(x)+S(y)+T(z)}
for some one-variable polynomials
R
,
S
,
T
{R,S,T}
, we have
|
f
(
A
,
B
,
C
)
|
≫
min
{
q
r
,
|
A
|
|
B
|
|
C
|
q
2
r
-
1
}
|f(A,B,C)|\gg\min\biggl{\{}q^{r},\frac{|A||B||C|}{q^{2r-1}}\bigg{\}}
for any
A
,
B
,
C
⊂
ℛ
{A,B,C\subset\mathcal{R}}
.
We also study the sum-product type problems over finite valuation ring
ℛ
{\mathcal{R}}
. More precisely, we show that for any
A
⊂
ℛ
{A\subset\mathcal{R}}
with
|
A
|
≫
q
r
-
1
3
{|A|\gg q^{r-\frac{1}{3}}}
then
max
{
|
A
A
|
,
|
A
d
+
A
d
|
}
{\max\{|AA|,|A^{d}+A^{d}|\}}
,
max
{
|
A
+
A
|
,
|
A
2
+
A
2
|
}
{\max\{|A+A|,|A^{2}+A^{2}|\}}
,
max
{
|
A
-
A
|
,
|
A
A
+
A
A
|
}
≫
|
A
|
2
3
q
r
3
{\max\{|A-A|,|AA+AA|\}\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}}
,
and
|
f
(
A
)
+
A
|
≫
|
A
|
2
3
q
r
3
{|f(A)+A|\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}}
for any one variable quadratic polynomial f.
Publisher
Walter de Gruyter GmbH
Subject
Applied Mathematics,General Mathematics
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