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We prove a new bound for multiple Gauss sums and, as an application, improve previous results in the Birch--Goldbach problem. Let $F_1, \\ldots, F_R \\in \\mathbb{Z}[x_1, \\ldots, x_s]$ be forms with differing degrees, with $D$ being the highest degree, and let $\\boldsymbol{F} = (F_1, \\ldots, F_R)$ be nonsingular. We prove that the system $\\boldsymbol{F}(\\boldsymbol{x})=\\mathbf{0}$ is solvable in primes provided that $s \\geq D^2 4^{D+2} R^5$.","authors_text":"Jianya Liu, Sizhe Xie","cross_cats":[],"headline":"A bound on multiple Gauss sums proves that nonsingular form systems have prime solutions when the number of variables meets or exceeds D squared times 4 to the D plus 2 times R to the fifth.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-04-03T10:38:32Z","title":"Multiple Gauss sums"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.03347","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-13T18:46:56.776255Z","id":"865f62c4-6b00-456e-8aad-a7e98e3d6473","model_set":{"reader":"grok-4.3"},"one_line_summary":"New bound on multiple Gauss sums improves the Birch-Goldbach result: nonsingular systems of R forms of max degree D in s variables have prime solutions when s ≥ D² 4^{D+2} R^5.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"A bound on multiple Gauss sums proves that nonsingular form systems have prime solutions when the number of variables meets or exceeds D squared times 4 to the D plus 2 times R to the fifth.","strongest_claim":"We prove that the system F(x)=0 is solvable in primes provided that s ≥ D² 4^{D+2} R^5, where F consists of R nonsingular forms of differing degrees with maximum degree D.","weakest_assumption":"The system of forms is nonsingular; the proof relies on this algebraic condition to control the singular series or major arcs in the analytic argument."}},"verdict_id":"865f62c4-6b00-456e-8aad-a7e98e3d6473"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a5bfecc21e0454fb1823ca047bcf49afc228393c67e32f42cefd568d2c55c4ce","target":"record","created_at":"2026-05-20T00:03:10Z","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":"0aa73b3a500a57e6f4ff7ea7b4872172029ab445ce414544126752e5521dac7b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-04-03T10:38:32Z","title_canon_sha256":"4e628f608e030f9b97f28822412875183c276145b12904b389819edc7b750c47"},"schema_version":"1.0","source":{"id":"2604.03347","kind":"arxiv","version":2}},"canonical_sha256":"bec3e4c1d4f53dbec2af546142f0c06551e6b6f6868e0ebb035e4c4442464e61","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bec3e4c1d4f53dbec2af546142f0c06551e6b6f6868e0ebb035e4c4442464e61","first_computed_at":"2026-05-20T00:03:10.454398Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:03:10.454398Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5qZnCWLp1lEubh4A1nyEWPUUAtXKM0CTPlXAVeO6vwqDz8IiBBDTW2isIGV6U80SAKhgRGut2sIjjBDF0LX0Cg==","signature_status":"signed_v1","signed_at":"2026-05-20T00:03:10.455225Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.03347","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a5bfecc21e0454fb1823ca047bcf49afc228393c67e32f42cefd568d2c55c4ce","sha256:a42c12f66e0ee5b81636399f350d77bfe17f4be004785683e9663d7d2d78c79f"],"state_sha256":"bd85a5df0e59cb49832eab7bf34d4b0b79774dd8a58ad489a0769e766de860ea"}