{"paper":{"title":"Multiple Gauss sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","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.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jianya Liu, Sizhe Xie","submitted_at":"2026-04-03T10:38:32Z","abstract_excerpt":"A multiple Gauss sum is a complete multiple exponential sum twisted by Dirichlet characters. 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 $s \\geq D^2 4^{D+2} R^5$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c656bfd670fc82f83cb035952694746defcbd137bc16b48c24715db7f6dfd4c5"},"source":{"id":"2604.03347","kind":"arxiv","version":2},"verdict":{"id":"865f62c4-6b00-456e-8aad-a7e98e3d6473","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T18:46:56.776255Z","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.","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","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.","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."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.03347/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}