{"paper":{"title":"A-Discriminants for Complex Exponents, and Counting Real Isotopy Types","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SC","math.CV"],"primary_cat":"math.AG","authors_text":"J. Maurice Rojas, Korben Rusek","submitted_at":"2016-12-11T19:22:40Z","abstract_excerpt":"We extend the definition of $\\mathcal{A}$-discriminant varieties, and Kapranov's parametrization of $\\mathcal{A}$-discriminant varieties, to complex exponents. As an application, we study the special case where $\\mathcal{A}$ is a fixed real $n\\times (n+3)$ matrix whose columns form the spectrum of an $n$-variate exponential sum $g$ with fixed sign vector for its coefficients: We prove that the number of possible isotopy types for the real zero set of $g$ is $O(n^2)$. The best previous upper bound was $2^{O(n^4)}$. Along the way, we also show that the singular loci of our generalized $\\mathcal{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.03458","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"}