The Ramanujan Journal

, Volume 47, Issue 1, pp 67–84

# Counting polynomial subset sums

• Jiyou Li
• Daqing Wan
Article

## 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.

## Keywords

Polynomial subset sums Inclusion–exclusion Character sums Subset sum problem Counting problems

## Mathematics Subject Classification

11T06 11T24 05A15 05A16

## Notes

### Acknowledgements

The authors wish to thank Professor Richard Stanley for his helpful suggestions.

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## Authors and Affiliations

• Jiyou Li
• 1
• 2
• Daqing Wan
• 3
1. 1.Department of MathematicsShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
3. 3.Department of MathematicsUniversity of CaliforniaIrvineUSA