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We then improve these bounds in the case of two variables polynomials of large degree to get a decomposition $P(x,y) = Q_1(x,y)^k+...+ Q_s(x,y)^k$ with $\\deg Q_i^k \\le \\deg P + k^3$ and $s$ that depends on $k$ and $\\ln (\\deg P)$."},"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":"1104.0472","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-04-04T07:01:42Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"a9e5221912e020749ffb00d857390529c7ac68fc09660eb0b12efb3251476663","abstract_canon_sha256":"2e23b5f40b3924fc62d9e2b10fdc26fcb6348c16074ed9f1af880d1f25957128"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:47.034620Z","signature_b64":"liUAH1ccLRnk7KyqzAUdZwMhrfwJYluWX80Dc4S/w0W5P0Qo+FHIDL5yS0s+/ThnR5dZ72BHoJVKplhQPJrYBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14a28fdf51ca062d03cc539c987a2ba04e050cce57865a5ab838307380f537a9","last_reissued_at":"2026-05-18T04:10:47.034140Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:47.034140Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Waring's problem for polynomials in two variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.NT","authors_text":"Arnaud Bodin, Mireille Car","submitted_at":"2011-04-04T07:01:42Z","abstract_excerpt":"We prove that all polynomials in several variables can be decomposed as the sums of $k$th powers: $P(x_1,...,x_n) = Q_1(x_1,...,x_n)^k+...+ Q_s(x_1,...,x_n)^k$, provided that elements of the base field are themselves sums of $k$th powers. We also give bounds for the number of terms $s$ and the degree of the $Q_i^k$. We then improve these bounds in the case of two variables polynomials of large degree to get a decomposition $P(x,y) = Q_1(x,y)^k+...+ Q_s(x,y)^k$ with $\\deg Q_i^k \\le \\deg P + k^3$ and $s$ that depends on $k$ and $\\ln (\\deg P)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.0472","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":"1104.0472","created_at":"2026-05-18T04:10:47.034209+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.0472v3","created_at":"2026-05-18T04:10:47.034209+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.0472","created_at":"2026-05-18T04:10:47.034209+00:00"},{"alias_kind":"pith_short_12","alias_value":"CSRI7X2RZIDC","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"CSRI7X2RZIDC2A6M","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"CSRI7X2R","created_at":"2026-05-18T12:26:26.731475+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/CSRI7X2RZIDC2A6MKOOJQ6RLUB","json":"https://pith.science/pith/CSRI7X2RZIDC2A6MKOOJQ6RLUB.json","graph_json":"https://pith.science/api/pith-number/CSRI7X2RZIDC2A6MKOOJQ6RLUB/graph.json","events_json":"https://pith.science/api/pith-number/CSRI7X2RZIDC2A6MKOOJQ6RLUB/events.json","paper":"https://pith.science/paper/CSRI7X2R"},"agent_actions":{"view_html":"https://pith.science/pith/CSRI7X2RZIDC2A6MKOOJQ6RLUB","download_json":"https://pith.science/pith/CSRI7X2RZIDC2A6MKOOJQ6RLUB.json","view_paper":"https://pith.science/paper/CSRI7X2R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.0472&json=true","fetch_graph":"https://pith.science/api/pith-number/CSRI7X2RZIDC2A6MKOOJQ6RLUB/graph.json","fetch_events":"https://pith.science/api/pith-number/CSRI7X2RZIDC2A6MKOOJQ6RLUB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CSRI7X2RZIDC2A6MKOOJQ6RLUB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CSRI7X2RZIDC2A6MKOOJQ6RLUB/action/storage_attestation","attest_author":"https://pith.science/pith/CSRI7X2RZIDC2A6MKOOJQ6RLUB/action/author_attestation","sign_citation":"https://pith.science/pith/CSRI7X2RZIDC2A6MKOOJQ6RLUB/action/citation_signature","submit_replication":"https://pith.science/pith/CSRI7X2RZIDC2A6MKOOJQ6RLUB/action/replication_record"}},"created_at":"2026-05-18T04:10:47.034209+00:00","updated_at":"2026-05-18T04:10:47.034209+00:00"}