Hilbert series of subspace arrangements |
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Authors: | Harm Derksen |
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Affiliation: | Department of Mathematics, University of Michigan, East Hall, 530 Church Street, 48109 Ann Arbor, MI, United States |
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Abstract: | The vanishing ideal I of a subspace arrangement V1∪V2∪?∪Vm⊆V is an intersection I1∩I2∩?∩Im of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J=I1I2?Im without any assumptions about the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. We will also apply our results to generalized principal component analysis (GPCA), a tool that is useful for computer vision and image processing. |
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Keywords: | Primary, 13D40 secondary, 13D02, 68T45 |
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