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Then, we evaluate $\\sum f(x_1,\\ldots,x_k)$, where the summation is taken over all pairwise distinct $x_1,\\ldots,x_k \\in G$. In particular, let $p^s$ be a power of an odd prime, $n$ a positive integer coprime with $p-1$, and $a_1,\\ldots,a_k$ integers such that $\\varphi(p^s)$ divides $a_1+\\cdots+a_k$ and $p-1$ does not divide $\\sum_{i \\in I}a_i$ for all non-empty proper subsets $I\\subseteq"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1411.2269","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-11-09T19:20:17Z","cross_cats_sorted":[],"title_canon_sha256":"05ad5d6eaf78a5e7f65f06b7518c3fd9c67bf2e4b35d52c9145823f39d04e7a1","abstract_canon_sha256":"3a7b211c8db07f6932b2e569f58c3f51f94d156308a2c0572e32684e895166a9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:28.714917Z","signature_b64":"P9ENDDl6TbyMJBehYK9/+EykRt5gW3WxbtS33gPueJ9xbrWu6DXbPQXQdgiYOOawMnx1/zk8mASSxKX/4ROdCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3da75aa07cabc2f546f062d948e75d57981a1a3779fa60165db6b14d4c2f855","last_reissued_at":"2026-05-18T00:44:28.714363Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:28.714363Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sums of Multivariate Polynomials in Finite Subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrea Marino, Paolo Leonetti","submitted_at":"2014-11-09T19:20:17Z","abstract_excerpt":"Let $R$ be a commutative ring, $f \\in R[X_1,\\ldots,X_k]$ a multivariate polynomial, and $G$ a finite subgroup of the group of units of $R$ satisfying a certain constraint, which always holds if $R$ is a field. Then, we evaluate $\\sum f(x_1,\\ldots,x_k)$, where the summation is taken over all pairwise distinct $x_1,\\ldots,x_k \\in G$. In particular, let $p^s$ be a power of an odd prime, $n$ a positive integer coprime with $p-1$, and $a_1,\\ldots,a_k$ integers such that $\\varphi(p^s)$ divides $a_1+\\cdots+a_k$ and $p-1$ does not divide $\\sum_{i \\in I}a_i$ for all non-empty proper subsets $I\\subseteq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.2269","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1411.2269","created_at":"2026-05-18T00:44:28.714445+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.2269v3","created_at":"2026-05-18T00:44:28.714445+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.2269","created_at":"2026-05-18T00:44:28.714445+00:00"},{"alias_kind":"pith_short_12","alias_value":"UPNHLKQHZK6C","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_16","alias_value":"UPNHLKQHZK6C6VDP","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_8","alias_value":"UPNHLKQH","created_at":"2026-05-18T12:28:52.271510+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UPNHLKQHZK6C6VDPAYWZJDTV2V","json":"https://pith.science/pith/UPNHLKQHZK6C6VDPAYWZJDTV2V.json","graph_json":"https://pith.science/api/pith-number/UPNHLKQHZK6C6VDPAYWZJDTV2V/graph.json","events_json":"https://pith.science/api/pith-number/UPNHLKQHZK6C6VDPAYWZJDTV2V/events.json","paper":"https://pith.science/paper/UPNHLKQH"},"agent_actions":{"view_html":"https://pith.science/pith/UPNHLKQHZK6C6VDPAYWZJDTV2V","download_json":"https://pith.science/pith/UPNHLKQHZK6C6VDPAYWZJDTV2V.json","view_paper":"https://pith.science/paper/UPNHLKQH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.2269&json=true","fetch_graph":"https://pith.science/api/pith-number/UPNHLKQHZK6C6VDPAYWZJDTV2V/graph.json","fetch_events":"https://pith.science/api/pith-number/UPNHLKQHZK6C6VDPAYWZJDTV2V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UPNHLKQHZK6C6VDPAYWZJDTV2V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UPNHLKQHZK6C6VDPAYWZJDTV2V/action/storage_attestation","attest_author":"https://pith.science/pith/UPNHLKQHZK6C6VDPAYWZJDTV2V/action/author_attestation","sign_citation":"https://pith.science/pith/UPNHLKQHZK6C6VDPAYWZJDTV2V/action/citation_signature","submit_replication":"https://pith.science/pith/UPNHLKQHZK6C6VDPAYWZJDTV2V/action/replication_record"}},"created_at":"2026-05-18T00:44:28.714445+00:00","updated_at":"2026-05-18T00:44:28.714445+00:00"}