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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1203.6498","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-03-29T12:04:57Z","cross_cats_sorted":[],"title_canon_sha256":"e01aa45a3b94ccf11869ac3052701709554592f3bf4cda6ae69433e3b8b0af8e","abstract_canon_sha256":"1dcaf40d855789cbff3ebad2f250ac6107b7cc88a1166c6f93aef9188af8fc03"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:14.098540Z","signature_b64":"ZsUVoc+epqDBHfElOg0haHgR8hnBLxxuZFnl08uO32DnWoORmaNqMUyEzqfmjVVPapwGR3EbnYn5GWQZBAwgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba56edf68037d8b6083f16fe50e0ff8aeb2c42f3996a8bae9513751f256d36d3","last_reissued_at":"2026-05-18T03:51:14.097619Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:14.097619Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1203.6498","created_at":"2026-05-18T03:51:14.097772+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.6498v3","created_at":"2026-05-18T03:51:14.097772+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.6498","created_at":"2026-05-18T03:51:14.097772+00:00"},{"alias_kind":"pith_short_12","alias_value":"XJLO35UAG7ML","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_16","alias_value":"XJLO35UAG7MLMCB7","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_8","alias_value":"XJLO35UA","created_at":"2026-05-18T12:27:27.928770+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2304.02415","citing_title":"Utilisation de l'aplatissement en g\\'eom\\'etrie de Berkovich","ref_index":8,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XJLO35UAG7MLMCB7C37FBYH7RL","json":"https://pith.science/pith/XJLO35UAG7MLMCB7C37FBYH7RL.json","graph_json":"https://pith.science/api/pith-number/XJLO35UAG7MLMCB7C37FBYH7RL/graph.json","events_json":"https://pith.science/api/pith-number/XJLO35UAG7MLMCB7C37FBYH7RL/events.json","paper":"https://pith.science/paper/XJLO35UA"},"agent_actions":{"view_html":"https://pith.science/pith/XJLO35UAG7MLMCB7C37FBYH7RL","download_json":"https://pith.science/pith/XJLO35UAG7MLMCB7C37FBYH7RL.json","view_paper":"https://pith.science/paper/XJLO35UA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.6498&json=true","fetch_graph":"https://pith.science/api/pith-number/XJLO35UAG7MLMCB7C37FBYH7RL/graph.json","fetch_events":"https://pith.science/api/pith-number/XJLO35UAG7MLMCB7C37FBYH7RL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XJLO35UAG7MLMCB7C37FBYH7RL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XJLO35UAG7MLMCB7C37FBYH7RL/action/storage_attestation","attest_author":"https://pith.science/pith/XJLO35UAG7MLMCB7C37FBYH7RL/action/author_attestation","sign_citation":"https://pith.science/pith/XJLO35UAG7MLMCB7C37FBYH7RL/action/citation_signature","submit_replication":"https://pith.science/pith/XJLO35UAG7MLMCB7C37FBYH7RL/action/replication_record"}},"created_at":"2026-05-18T03:51:14.097772+00:00","updated_at":"2026-05-18T03:51:14.097772+00:00"}