{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:U6URA6XVVCJXSUHKSLK22ABUMP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b5a05cd3d3ad6467bfd35414ed815bd8503afbc710b4a0fbe73756bcb8a87769","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-10-04T16:28:25Z","title_canon_sha256":"86007a8fc3f645dc9466695344cf5671ad753fccaa0cc017f42d2520c41953d3"},"schema_version":"1.0","source":{"id":"1110.0745","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.0745","created_at":"2026-05-18T04:11:41Z"},{"alias_kind":"arxiv_version","alias_value":"1110.0745v1","created_at":"2026-05-18T04:11:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.0745","created_at":"2026-05-18T04:11:41Z"},{"alias_kind":"pith_short_12","alias_value":"U6URA6XVVCJX","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"U6URA6XVVCJXSUHK","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"U6URA6XV","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:f3bed147c01c382f2382abcab022db34e95539fc53323425955dc13d27e6ddeb","target":"graph","created_at":"2026-05-18T04:11:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In the polynomial ring $T=k[y_1,...,y_n]$, with $n>1$, we bound the multiplicity of homogeneous radical ideals $I\\subset (y_1^{a_1},...,y_n^{a_n})$ such that $T/I$ is a graded $k$-algebra with Krull dimension one. As a consequence we solve the Waring Problem for all monomials, i.e. we compute the minimal number of linear forms needed to write a monomial as a sum of powers of these linear forms. Moreover, we give an explicit description of a sum of powers decomposition for monomials. We also produce new bounds for the Waring rank of polynomials which are a sum of pairwise coprime monomials.","authors_text":"Anthony V. Geramita, Enrico Carlini, Maria Virginia Catalisano","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-10-04T16:28:25Z","title":"The Solution to Waring's Problem for Monomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.0745","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:bc88f4e697b154e41eae6f9f548f272358b1ae05387c3b378f835caf2f9c6453","target":"record","created_at":"2026-05-18T04:11:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b5a05cd3d3ad6467bfd35414ed815bd8503afbc710b4a0fbe73756bcb8a87769","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-10-04T16:28:25Z","title_canon_sha256":"86007a8fc3f645dc9466695344cf5671ad753fccaa0cc017f42d2520c41953d3"},"schema_version":"1.0","source":{"id":"1110.0745","kind":"arxiv","version":1}},"canonical_sha256":"a7a9107af5a8937950ea92d5ad003463cac23bc903421ea5345a993ced01e42f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a7a9107af5a8937950ea92d5ad003463cac23bc903421ea5345a993ced01e42f","first_computed_at":"2026-05-18T04:11:41.108879Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:11:41.108879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7/KS0l/jwsmVUUsGkbgAP46MIsqeA4rFshFe6gzHPtf94Fclv4Vt+1Zio6Go1exj8Buhk/WWwnpZwGsu9JKGDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:11:41.109330Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.0745","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bc88f4e697b154e41eae6f9f548f272358b1ae05387c3b378f835caf2f9c6453","sha256:f3bed147c01c382f2382abcab022db34e95539fc53323425955dc13d27e6ddeb"],"state_sha256":"852b716c7ccf8629302a22ad45c44fbbb96a677d914178a99619b3dd9d484e6f"}