{"paper":{"title":"$L^p$-norms and Mahler's measure of polynomials on the $n$-dimensional torus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Andreas Defant, Mieczys{\\l}aw Masty{\\l}o","submitted_at":"2015-08-23T01:59:15Z","abstract_excerpt":"We prove Nikol'skii type inequalities which for polynomials on the $n$-dimensional torus $\\mathbb{T}^n$ relate the $L^p$-with the $L^q$-norm (with respect to the normalized Lebesgue measure and $0 <p <q < \\infty$). Among other things we show that $C=\\sqrt{q/p}$ is the best constant such that $\\|P\\|_{L^q}\\leq C^{\\text{deg}(P)} \\|P\\|_{L^p}$ for all homogeneous polynomials $P$ on $\\mathbb{T}^n$. We also prove an exact inequality between the $L^p$-norm of a polynomial $P$ on $\\mathbb{T}^n$ and its Mahler measure $M(P)$, which is the geometric mean of $|P|$ with respect to the normalized Lebesgue m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.05556","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"}