{"paper":{"title":"Jacobian-squared function-germs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Takashi Nishimura","submitted_at":"2018-06-19T13:35:43Z","abstract_excerpt":"In this paper, it is shown that, for any equidimensional $C^\\infty$ map-germ $f: (\\mathbb{R}^n,0)\\to (\\mathbb{R}^n,0)$, the map-germ $F: (\\mathbb{R}^n, 0) \\to \\mathbb{R}^n\\times\\mathbb{R}^{\\ell}$ defined by $F(x)=\\left(f(x), \\mu_1(x){|Jf|^2(x)}, \\cdots, \\mu_\\ell(x){|Jf|^2(x)}\\right)$ is always a frontal; where $\\mu_i$ is a $C^\\infty$ function-germ and $|Jf|$ is the Jacobian-determinant of $f$. Moreover, it is also shown that when the multiplicity of $f$ is less than or equal to $3$, any frontal constructed from $f$ must be $\\mathcal{A}$-equivalent to a frontal $F$ of the above form."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.10208","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":""},"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"}