{"paper":{"title":"Canonical heights and preperiodic points for weighted homogeneous families of polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Patrick Ingram","submitted_at":"2015-10-29T18:19:28Z","abstract_excerpt":"A family $f_t(z)$ of polynomials over a number field $K$ will be called \\emph{weighted homogeneous} if and only if $f_t(z)=F(z^e, t)$ for some binary homogeneous form $F(X, Y)$ and some integer $e\\geq 2$. For example, the family $z^d+t$ is weighted homogeneous. We prove a lower bound on the canonical height, of the form \\[\\hat{h}_{f_t}(z)\\geq \\epsilon \\max\\{h_{\\mathsf{M}_d}(f_t), \\log|\\operatorname{Norm}\\mathfrak{R}_{f_t}|\\},\\] for values $z\\in K$ which are not preperiodic for $f_t$. Here $\\epsilon$ depends only on the number of places at which $f_t$ has bad reduction. For suitably generic mor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08807","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"}