Counting polynomial subset sums
Name
11139_2018_20_ReferencePDF.pdf
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215.82 KB
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74f1eb472692826312f7002176b1879b
Author(s)
Li, Jiyou
Wan, Daqing
Date Issued
July 11, 2018
Publisher
Springer US
Version
Author's final manuscript
Abstract
Abstract
Let D be a subset of a finite commutative ring R with identity. Let
$$f(x)\in R[x]$$
f
(
x
)
∈
R
[
x
]
be a polynomial of degree d. For a nonnegative integer k, we study the number
$$N_f(D,k,b)$$
N
f
(
D
,
k
,
b
)
of k-subsets S in D such that
$$\begin{aligned} \sum _{x\in S} f(x)=b. \end{aligned}$$
∑
x
∈
S
f
(
x
)
=
b
.
In this paper, we establish several bounds for the difference between
$$N_f(D,k, b)$$
N
f
(
D
,
k
,
b
)
and the expected main term
$$\frac{1}{|R|}{|D|\atopwithdelims ()k}$$
1
|
R
|
|
D
|
k
, depending on the nature of the finite ring R and f. For
$$R=\mathbb {Z}_n$$
R
=
Z
n
, let
$$p=p(n)$$
p
=
p
(
n
)
be the smallest prime divisor of n,
$$|D|=n-c \ge C_dn p^{-\frac{1}{d}}\,+\,c$$
|
D
|
=
n
-
c
≥
C
d
n
p
-
1
d
+
c
and
$$f(x)=a_dx^d +\cdots +a_0\in \mathbb {Z}[x]$$
f
(
x
)
=
a
d
x
d
+
⋯
+
a
0
∈
Z
[
x
]
with
$$(a_d, \ldots , a_1, n)=1$$
(
a
d
,
…
,
a
1
,
n
)
=
1
. Then
$$\begin{aligned} \left| N_f(D, k, b)-\frac{1}{n}{n-c \atopwithdelims ()k}\right| \le {\delta (n)(n-c)+(1-\delta (n))\left( C_dnp^{-\frac{1}{d}}+c\right) +k-1\atopwithdelims ()k}, \end{aligned}$$
N
f
(
D
,
k
,
b
)
-
1
n
n
-
c
k
≤
δ
(
n
)
(
n
-
c
)
+
(
1
-
δ
(
n
)
)
C
d
n
p
-
1
d
+
c
+
k
-
1
k
,
answering an open question raised by Stanley (Enumerative combinatorics, 1997) in a general setting, where
$$\delta (n)=\sum _{i\mid n, \mu (i)=-1}\frac{1}{i}$$
δ
(
n
)
=
∑
i
∣
n
,
μ
(
i
)
=
-
1
1
i
and
$$C_d=e^{1.85d}$$
C
d
=
e
1.85
d
. Furthermore, if n is a prime power, then
$$\delta (n) =1/p$$
δ
(
n
)
=
1
/
p
and one can take
$$C_d=4.41$$
C
d
=
4.41
. Similar and stronger bounds are given for two more cases. The first one is when
$$R=\mathbb {F}_q$$
R
=
F
q
, a q-element finite field of characteristic p and f(x) is general. The second one is essentially the well-known subset sum problem over an arbitrary finite abelian group. These bounds extend several previous results.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s11139-018-0020-0
