{"paper":{"title":"What is the probability that a random integral quadratic form in $n$ variables is isotropic?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"John Cremona, Manjul Bhargava, Tom Fisher","submitted_at":"2013-11-21T20:31:12Z","abstract_excerpt":"We show that the density of quadratic forms in $n$ variables over ${\\mathbb Z}_p$ that are isotropic is a rational function in $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. As a consequence, for each $n$, we determine the probability that a random integral quadratic form in $n$ variables is isotropic. In particular, we show that the probability that a random integral quaternary quadratic form is isotropic is $\\approx 97.0\\%$, in the case where the coefficients of the quadratic form are independently and uniformly distributed in the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.5543","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"}