Counting polynomial subset sums

TL;DR

This paper derives asymptotic formulas for counting subsets of finite rings and groups that satisfy polynomial sum conditions, extending subset sum analysis to algebraic structures and polynomial functions.

Contribution

It provides new asymptotic formulas for subset sum counts over finite rings and groups, including cases with polynomial functions, partially answering an open question by Stanley.

Findings

Asymptotic formulas for polynomial subset sums over _q and _n

Explicit bounds for subset sum counts in finite abelian groups

New proof for the subset sum count in finite groups

Abstract

Let $D$ be a subset of a finite commutative ring $R$ with identity. Let $f(x)\in R[x]$ be a polynomial of positive degree $d$. For integer $0\leq k \leq |D|$, we study the number $N_f(D,k,b)$ of $k$-subsets $S\subseteq D$ such that \begin{align*} \sum_{x\in S} f(x)=b. \end{align*} In this paper, we establish several asymptotic formulas for $N_f(D,k, b)$, depending on the nature of the ring $R$ and $f$. For $R=\mathbb{Z}_n$, let $p=p(n)$ be the smallest prime divisor of $n$, $|D|=n-c \geq C_dn p^{-\frac 1d }+c$ and $f(x)=a_dx^d +\cdots +a_0\in \mathbb{Z}[x]$ with $(a_d, \dots, a_1, n)=1$. Then $$\left| N_f(D, k, b)-\frac{1}{n}{n-c \choose k}\right|\leq {\delta(n)(n-c)+(1-\delta(n))(C_dnp^{-\frac 1d}+c)+k-1\choose k},$$ partially answering an open question raised by Stanley \cite{St}, 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 $C_d=4.41$. For $R=\mathbb{F}_q$ of characteristic $p$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $d$ not divisible by $p$ and $D\subseteq \mathbb{F}_q$ with $|D|=q-c\geq (d-1)\sqrt{q}+c$. Then $$\left| N_f(D, k, b)-\frac{1}{q}{q-c \choose k}\right|\leq {\frac{q-c}{p}+\frac {p-1}{p}((d-1)q^{\frac 12}+c)+k-1 \choose k}.$$ If $f(x)=ax+b$, then this problem is precisely the well-known subset sum problem over a finite abelian group. Let $G$ be a finite abelian group and let $D\subseteq G$ with $|D|=|G|-c\geq c$. Then $$\left| N_x(D, k, b)-\frac{1}{|G|}{|G|-c \choose k}\right|\leq {c + (|G|-2c)\delta(e(G))+k-1 \choose k},$$ where $e(G)$ is the exponent of $G$ and $\delta(n)=\sum_{i\mid n, \mu(i)=-1}\frac 1 i$. In particular, we give a new short proof for the explicit counting formula for the case $D=G$.