Counting polynomial subset sums

Abstract

Let D be a subset of a finite commutative ring R with identity. Let $f(x)\in R[x]$ be a polynomial of degree d. For a nonnegative integer k, we study the number $N_f(D,k,b)$ of k-subsets S in D such that

$$\begin{aligned} \sum _{x\in S} f(x)=b. \end{aligned}$$

In this paper, we establish several bounds for the difference between $N_f(D,k, b)$ and the expected main term $\frac{1}{|R|}{|D|\atopwithdelims ()k}$, depending on the nature of the finite ring R and f. For $R=\mathbb {Z}_n$, let $p=p(n)$ be the smallest prime divisor of n, $|D|=n-c \ge C_dn p^{-\frac{1}{d}}\,+\,c$ and $f(x)=a_dx^d +\cdots +a_0\in \mathbb {Z}[x]$ with $(a_d, \ldots , a_1, n)=1$. Then

$$\begin{aligned} \left| N_f(D, k, b)-\frac{1}{n}{n-c \atopwithdelims ()k}\right| \le {\delta (n)(n-c)+(1-\delta (n))\left( C_dnp^{-\frac{1}{d}}+c\right) +k-1\atopwithdelims ()k}, \end{aligned}$$

answering an open question raised by Stanley (Enumerative combinatorics, 1997) in a general setting, where $\delta (n)=\sum _{i\mid n, \mu (i)=-1}\frac{1}{i}$ and $C_d=e^{1.85d}$. Furthermore, if n is a prime power, then $\delta (n) =1/p$ and one can take $C_d=4.41$. Similar and stronger bounds are given for two more cases. The first one is when $R=\mathbb {F}_q$, 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.