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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 $p<k\\leq q$. 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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 $p<k\\leq q$. 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