{"paper":{"title":"Vanishing products of one-forms and critical points of master functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Alexander Varchenko, Daniel C. Cohen, Graham Denham, Michael Falk","submitted_at":"2010-10-18T21:35:15Z","abstract_excerpt":"Let \\A be an affine hyperplane arrangement in $\\C^\\ell$ with complement $U$. Let $f_1, \\..., f_n$ be linear polynomials defining the hyperplanes of \\A, and $A^\\cdot$ the algebra of differential forms generated by the 1-forms $d \\log f_1, \\..., d \\log f_n$. To each $l \\in \\C^n$ we associate the master function $\\Phi=\\Phi_l = \\prod_{i=1}^n f_i^{l_i}$ on $U$ and the closed logarithmic 1-form $\\omega= d \\log \\Phi$. We assume $\\omega$ is an element of a rational linear subspace $D$ of $A^1$ of dimension $q>1$ such that the multiplication map $\\bigwedge^k(D) \\to A^k$ is zero for $p<k\\leq q$. With th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.3743","kind":"arxiv","version":3},"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"}