{"paper":{"title":"Higher dimensional Lemniscates: the geometry of $r$ particles in $n$-space with logarithmic potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.MG"],"primary_cat":"math.AG","authors_text":"Antonio Jose Di Scala (Politecnico di Torino), Fabrizio Catanese (Universitaet Bayreuth), Ingrid Bauer","submitted_at":"2015-06-05T13:59:22Z","abstract_excerpt":"We prove some basic theorems concerning lemniscate configurations in an Euclidean space of dimension $ n \\geq 3$. Lemniscates are defined as follows. Given m points $w_j $ in $\\mathbb R^n$, consider the function $F(x)$ which is the product of the distances $ |x-w_j|$: the singular level sets of the function $F$ are called lemniscates. We show via complex analysis that the critical points of $F$ have Hessian of positivity at least $(n-1)$. This implies that, if $F$ is a Morse function, then $F$ has only local minima and saddle points with negativity 1. The critical points lie in the convex span"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.01919","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"}