{"paper":{"title":"Imaginary projections of polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Thorsten J\\\"orgens, Thorsten Theobald, Timo de Wolff","submitted_at":"2016-02-05T13:03:37Z","abstract_excerpt":"We introduce the imaginary projection of a multivariate polynomial $f \\in \\mathbb{C}[\\mathbf{z}]$ as the projection of the variety of $f$ onto its imaginary part, $\\mathcal{I}(f) \\ = \\ \\{\\text{Im}(\\mathbf{z}) \\, : \\, \\mathbf{z} \\in \\mathcal{V}(f) \\}$. Since a polynomial $f$ is stable if and only if $\\mathcal{I}(f) \\cap \\mathbb{R}_{>0}^n \\ = \\ \\emptyset$, the notion offers a novel geometric view underlying stability questions of polynomials.\n  We show that the connected components of the complement of the closure of the imaginary projections are convex, thus opening a central connection to the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.02008","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"}