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We show that if $f$ satisfies the Mikhailov - Gindikin condition then \\begin{itemize} \\item[(i)] $\\text{Volume}\\ G^f(r) \\asymp r^\\theta (\\ln r)^k$ \\item[(ii)] $\\text{Card}\\left(G^f(r) \\cap \\overset{o}{\\ \\Z^n}\\right) \\asymp r^{\\theta'}(\\ln r)^{k'}$, as $r\\to \\infty$, \\end{itemize} where the exponents $\\theta,\\ k,\\ \\theta',\\ k'$ are determined explicitly in terms of the Newton polyhedra of $f$. \\\\ \\indent Moreover, the polynomial maps satisfy the Mikhailov - Gin"},"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":"1502.06091","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.AG","submitted_at":"2015-02-21T10:15:48Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"2c61d0ed455543c777a258ee2815072c30716e0b59c0a177c621b367fe99de5a","abstract_canon_sha256":"bf18ef204432766cdc1bc6103adbfe5675cd680076292b371e3359ddf96cc50d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:38.088141Z","signature_b64":"JCEYE8N1/xKf+Lhj3QSnfgHLx1JcTGBwrNTrhrOjMjfDwShFyc7t6CtJfw+08VYoUobtk2sFM8pijI2Dcq91DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df53051e6397740ceb3798945a7e6a9d1414a966c97b48d229e577b06a7c26b8","last_reissued_at":"2026-05-18T02:26:38.087701Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:38.087701Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the volume and the number of lattice of some semialgebraic sets","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AG","authors_text":"Ha Huy Vui, Tran Gia Loc","submitted_at":"2015-02-21T10:15:48Z","abstract_excerpt":"Let $f = (f_1,\\ldots,f_m) : \\R^n \\longrightarrow \\R^m$ be a polynomial map; $G^f(r) = \\{x\\in\\R^n : |f_i(x)| \\leq r,\\ i =1,\\ldots, m\\}$. We show that if $f$ satisfies the Mikhailov - Gindikin condition then \\begin{itemize} \\item[(i)] $\\text{Volume}\\ G^f(r) \\asymp r^\\theta (\\ln r)^k$ \\item[(ii)] $\\text{Card}\\left(G^f(r) \\cap \\overset{o}{\\ \\Z^n}\\right) \\asymp r^{\\theta'}(\\ln r)^{k'}$, as $r\\to \\infty$, \\end{itemize} where the exponents $\\theta,\\ k,\\ \\theta',\\ k'$ are determined explicitly in terms of the Newton polyhedra of $f$. \\\\ \\indent Moreover, the polynomial maps satisfy the Mikhailov - Gin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.06091","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1502.06091","created_at":"2026-05-18T02:26:38.087767+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.06091v1","created_at":"2026-05-18T02:26:38.087767+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.06091","created_at":"2026-05-18T02:26:38.087767+00:00"},{"alias_kind":"pith_short_12","alias_value":"35JQKHTDS52A","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_16","alias_value":"35JQKHTDS52AZ2ZX","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_8","alias_value":"35JQKHTD","created_at":"2026-05-18T12:29:02.477457+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/35JQKHTDS52AZ2ZXTCKFU7TKTU","json":"https://pith.science/pith/35JQKHTDS52AZ2ZXTCKFU7TKTU.json","graph_json":"https://pith.science/api/pith-number/35JQKHTDS52AZ2ZXTCKFU7TKTU/graph.json","events_json":"https://pith.science/api/pith-number/35JQKHTDS52AZ2ZXTCKFU7TKTU/events.json","paper":"https://pith.science/paper/35JQKHTD"},"agent_actions":{"view_html":"https://pith.science/pith/35JQKHTDS52AZ2ZXTCKFU7TKTU","download_json":"https://pith.science/pith/35JQKHTDS52AZ2ZXTCKFU7TKTU.json","view_paper":"https://pith.science/paper/35JQKHTD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.06091&json=true","fetch_graph":"https://pith.science/api/pith-number/35JQKHTDS52AZ2ZXTCKFU7TKTU/graph.json","fetch_events":"https://pith.science/api/pith-number/35JQKHTDS52AZ2ZXTCKFU7TKTU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/35JQKHTDS52AZ2ZXTCKFU7TKTU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/35JQKHTDS52AZ2ZXTCKFU7TKTU/action/storage_attestation","attest_author":"https://pith.science/pith/35JQKHTDS52AZ2ZXTCKFU7TKTU/action/author_attestation","sign_citation":"https://pith.science/pith/35JQKHTDS52AZ2ZXTCKFU7TKTU/action/citation_signature","submit_replication":"https://pith.science/pith/35JQKHTDS52AZ2ZXTCKFU7TKTU/action/replication_record"}},"created_at":"2026-05-18T02:26:38.087767+00:00","updated_at":"2026-05-18T02:26:38.087767+00:00"}