{"paper":{"title":"A Note on the Maximum Number of Zeros of $r(z) - \\bar{z}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"J\\\"org Liesen, Olivier S\\`ete, Robert Luce","submitted_at":"2014-10-01T07:52:07Z","abstract_excerpt":"An important theorem of Khavinson & Neumann (Proc. Amer. Math. Soc. 134(4), 2006) states that the complex harmonic function $r(z) - \\bar{z}$, where $r$ is a rational function of degree $n \\geq 2$, has at most $5 (n - 1)$ zeros. In this note we resolve a slight inaccuracy in their proof and in addition we show that for certain functions of the form $r(z) - \\bar{z}$ no more than $5 (n - 1) - 1$ zeros can occur. Moreover, we show that $r(z) - \\bar{z}$ is regular, if it has the maximal number of zeros."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.0134","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"}