{"paper":{"title":"Polynomial estimates, exponential curves and Diophantine approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Dan Coman, Evgeny A. Poletsky","submitted_at":"2010-09-22T16:21:03Z","abstract_excerpt":"Let $\\alpha\\in(0,1)\\setminus{\\Bbb Q}$ and $K=\\{(e^z,e^{\\alpha z}):\\,|z|\\leq1\\}\\subset{\\Bbb C}^2$. If $P$ is a polynomial of degree $n$ in ${\\Bbb C}^2$, normalized by $\\|P\\|_K=1$, we obtain sharp estimates for $\\|P\\|_{\\Delta^2}$ in terms of $n$, where $\\Delta^2$ is the closed unit bidisk. For most $\\alpha$, we show that $\\sup_P\\|P\\|_{\\Delta^2}\\leq\\exp(Cn^2\\log n)$. However, for $\\alpha$ in a subset ${\\mathcal S}$ of the Liouville numbers, $\\sup_P\\|P\\|_{\\Delta^2}$ has bigger order of growth. We give a precise characterization of the set ${\\mathcal S}$ and study its properties."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4408","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"}