{"paper":{"title":"Dehn-Sommerville from Gauss-Bonnet","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Oliver Knill","submitted_at":"2019-05-13T02:18:03Z","abstract_excerpt":"We give a zero curvature proof of Dehn-Sommerville for finite simple graphs. It uses a parametrized Gauss-Bonnet formula telling that the curvature of the valuation G to f_G(t)=1+f0 t + ... + fd t^(d+1) defined by the f-vector of G is the anti-derivative F of f evaluated on the unit sphere S(x). Gauss Bonnet is then parametrized, f_G(t) = sum_x F_(S(x))(t), and holds for all simplicial complexes G. The Gauss-Bonnet formula chi(G)=sum_x K(x) for Euler characteristic chi(G) is the special case t=-1. Dehn-Sommerville is equivalent to the reflection symmetry f_G(t)+(-1)^d f_G(-1-t)=0 which is equi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.04831","kind":"arxiv","version":1},"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"}