{"paper":{"title":"Singularities of solutions to quadratic vector equations on complex upper half-plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP","math.SP"],"primary_cat":"math.PR","authors_text":"Laszlo Erdos, Oskari Ajanki, Torben Kr\\\"uger","submitted_at":"2015-12-06T17:56:21Z","abstract_excerpt":"Let $ S $ be a positivity preserving symmetric linear operator acting on bounded functions. The nonlinear equation $ -\\frac{1}{m}=z+Sm $ with a parameter $ z $ in the complex upper half-plane $ \\mathbb{H} $ has a unique solution $ m $ with values in $ \\mathbb{H} $. We show that the $ z $-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures $ v $ on $ \\mathbb{R} $. Under suitable conditions on $ S $, we show that $ v $ has a real analytic density apart from finitely many algebraic singularities of degree at most three.\n  Our motivation c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03703","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"}