{"paper":{"title":"Espaces de Berkovich, polytopes, squelettes et th\\'eorie des mod\\`eles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Antoine Ducros","submitted_at":"2012-03-29T12:04:57Z","abstract_excerpt":"Let $X$ be an analytic space over a non-Archimedean, complete field $k$ and let $(f_1,..., f_n)$ be a family of invertible functions on $X$. Let $\\phi$ the morphism $X\\to G_m^n$ induced by the $f_i$'s, and let $t$ be the map $X\\to (R^*_+)^n$ induced by the norms of the $f_i$'s. Let us recall two results.\n  1) The compact set $t(X)$ is a polytope of the $R$-vector space $(R^*_+)^n$ (we use the multiplicative notation) ; this is due to Berkovich in the locally algebraic case, and has been extended to the general case by the author.\n  2) If moreover $X$ is Hausdorff and $n$-dimensional, then the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6498","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"}