{"paper":{"title":"Many-point tropical relaxation and the Monge--Amp\\`ere equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.MG"],"primary_cat":"math.AP","authors_text":"Ernesto Lupercio, Higinio Serrano, Mikhail Shkolnikov, Nikita Kalinin","submitted_at":"2026-07-28T15:40:19Z","abstract_excerpt":"We construct an incidence-driven tropical approximation of the planar Aleksandrov Monge--Amp\\`ere equation. Let $\\Omega\\subset\\mathbb R^2$ be a bounded open convex domain, let $K\\Subset\\Omega$, and let $F_N=G_{P_N}0_\\Omega$ be the minimal nonnegative concave tropical series with integral slopes, zero boundary values, and corner locus containing an $N$-point set $P_N\\subset K$. For universally generic configurations whose empirical measures converge to $\\mu$, $N^{-1/2}F_N\\longrightarrow F_{\\mu,\\Omega}$ uniformly on $\\overline\\Omega$, where $F_{\\mu,\\Omega}$ is the unique continuous concave Aleks"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25878","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.25878/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}