{"paper":{"title":"Schanuel's theorem for heights defined via extension fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christopher Frei, Martin Widmer","submitted_at":"2012-08-23T15:49:51Z","abstract_excerpt":"Let $k$ be a number field, let $\\theta$ be a nonzero algebraic number, and let $H(\\cdot)$ be the Weil height on the algebraic numbers. In response to a question by T. Loher and D. W. Masser, we prove an asymptotic formula for the number of $\\alpha \\in k$ with $H(\\alpha \\theta)\\leq X$.\n  We also prove an asymptotic counting result for a new class of height functions defined via extension fields of $k$. This provides a conceptual framework for Loher and Masser's problem and generalizations thereof.\n  Moreover, we analyze the leading constant in our asymptotic formula for Loher and Masser's probl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4786","kind":"arxiv","version":2},"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"}