{"paper":{"title":"Sum of Squares Conjecture: the Monomial Case in $\\mathbb{C}^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CV","authors_text":"Dusty Grundmeier, Jennifer Brooks","submitted_at":"2021-07-30T16:35:18Z","abstract_excerpt":"The goal of this article is to prove the Sum of Squares Conjecture for real polynomials $r(z,\\bar{z})$ on $\\mathbb{C}^3$ with diagonal coefficient matrix. This conjecture describes the possible values for the rank of $r(z,\\bar{z}) \\|z\\|^2$ under the hypothesis that $r(z,\\bar{z})\\|z\\|^2=\\|h(z)\\|^2$ for some holomorphic polynomial mapping $h$. Our approach is to connect this problem to the degree estimates problem for proper holomorphic monomial mappings from the unit ball in $\\mathbb{C}^2$ to the unit ball in $\\mathbb{C}^k$. D'Angelo, Kos, and Riehl proved the sharp degree estimates theorem in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.14739","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/2107.14739/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"}