Counting polynomial subset sums

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.