{"paper":{"title":"Tropical resolutions of configuration hypersurfaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Any irreducible configuration hypersurface has its singularities resolved by normalizing the Nash blow-up and then taking a tropical compactification.","cross_cats":["math-ph","math.AC","math.MP"],"primary_cat":"math.AG","authors_text":"Daniel Bath, Graham Denham, Mathias Schulze, Uli Walther","submitted_at":"2026-05-05T19:37:09Z","abstract_excerpt":"Configuration polynomials generalize the Kirchhoff polynomial of a graph, as well as the Symanzik polynomials that appear in the denominators of Feynman integrands. The configuration hypersurfaces cut out by such polynomials are typically highly singular, which poses a challenge for the evaluation of Feynman integrals even in simplified settings.\n  In this paper, we provide a two-step recipe for a resolution of singularities of any irreducible configuration hypersurface. We first consider the normalization of the Nash blow-up, which we identify with an incidence variety introduced by Bloch. Th"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We provide a two-step recipe for a resolution of singularities of any irreducible configuration hypersurface. We first consider the normalization of the Nash blow-up, which we identify with an incidence variety introduced by Bloch. ... The latter then a smooth, tropical compactification, using work of Tevelev.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The configuration hypersurface is irreducible; the normalization of the Nash blow-up coincides with Bloch's incidence variety; Tevelev's tropical compactification theorem applies directly to the closure of the smooth torus subvariety; the bipermutohedral matroid combinatorics of Ardila-Denham-Huh yields an explicit smooth model.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A explicit two-step resolution of singularities for configuration hypersurfaces is constructed via normalized Nash blow-ups identified with Bloch incidence varieties followed by smooth tropical compactifications using bipermutohedral matroid combinatorics.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Any irreducible configuration hypersurface has its singularities resolved by normalizing the Nash blow-up and then taking a tropical compactification.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a96badefbdc517be9b90ecab8252c4c89432a46f1c375aeb1e3d36b69be7827c"},"source":{"id":"2605.04252","kind":"arxiv","version":2},"verdict":{"id":"2b5cf20d-365d-41db-a2e2-41c656f268ec","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T17:27:13.836462Z","strongest_claim":"We provide a two-step recipe for a resolution of singularities of any irreducible configuration hypersurface. We first consider the normalization of the Nash blow-up, which we identify with an incidence variety introduced by Bloch. ... The latter then a smooth, tropical compactification, using work of Tevelev.","one_line_summary":"A explicit two-step resolution of singularities for configuration hypersurfaces is constructed via normalized Nash blow-ups identified with Bloch incidence varieties followed by smooth tropical compactifications using bipermutohedral matroid combinatorics.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The configuration hypersurface is irreducible; the normalization of the Nash blow-up coincides with Bloch's incidence variety; Tevelev's tropical compactification theorem applies directly to the closure of the smooth torus subvariety; the bipermutohedral matroid combinatorics of Ardila-Denham-Huh yields an explicit smooth model.","pith_extraction_headline":"Any irreducible configuration hypersurface has its singularities resolved by normalizing the Nash blow-up and then taking a tropical compactification."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04252/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T12:35:59.475620Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T23:31:20.933149Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:40:39.134465Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"450d63ed8cb57ba3fbb1be3161311233f64ee507b32c04e64fcdbbc770310234"},"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"}