The Ramanujan Journal

, Volume 47, Issue 1, pp 67–84 | Cite as

Counting polynomial subset sums

  • Jiyou LiEmail author
  • Daqing Wan


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.


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

Mathematics Subject Classification

11T06 11T24 05A15 05A16 



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


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Authors and Affiliations

  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

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