{"paper":{"title":"Polynomial maps with invertible sums of Jacobian matrices and of directional Derivatives","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Hongbo Guo, Michiel de Bondt, Xiankun Du, Xiaosong Sun","submitted_at":"2011-06-04T06:46:45Z","abstract_excerpt":"Let $F: C^n \\rightarrow C^m$ be a polynomial map with $degF=d \\geq 2$. We prove that $F$ is invertible if $m = n$ and $\\sum^{d-1}_{i=1} JF(\\alpha_i)$ is invertible for all $i$, which is trivially the case for invertible quadratic maps. More generally, we prove that for affine lines $L = \\{\\beta + \\mu \\gamma | \\mu \\in C\\} \\subseteq C^n$ ($\\gamma \\ne 0$), $F|_L$ is linearly rectifiable, if and only if $\\sum^{d-1}_{i=1} JF(\\alpha_i) \\cdot \\gamma \\ne 0$ for all $\\alpha_i \\in L$. This appears to be the case for all affine lines $L$ when $F$ is injective and $d \\le 3$. We also prove that if $m = n$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.0792","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"}