{"paper":{"title":"Counting polynomial subset sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Daqing Wan, Jiyou Li","submitted_at":"2015-07-22T20:15:32Z","abstract_excerpt":"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\n  \\begin{align*}\n  \\sum_{x\\in S} f(x)=b.\n  \\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 $f$.\n  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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06329","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}