{"paper":{"title":"Regularity of roots of polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CA","authors_text":"Adam Parusinski, Armin Rainer","submitted_at":"2013-09-09T13:29:06Z","abstract_excerpt":"We show that smooth curves of monic complex polynomials $P_a (Z)=Z^n+\\sum_{j=1}^n a_j Z^{n-j}$, $a_j : I \\to \\mathbb C$ with $I \\subset \\mathbb R$ a compact interval, have absolutely continuous roots in a uniform way. More precisely, there exists a positive integer $k$ and a rational number $p >1$, both depending only on the degree $n$, such that if $a_j \\in C^{k}$ then any continuous choice of roots of $P_a$ is absolutely continuous with derivatives in $L^q$ for all $1 \\le q < p$, in a uniform way with respect to $\\max_j\\|a_j\\|_{C^k}$. The uniformity allows us to deduce also a multiparameter "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.2151","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"}