Hilbert series of subspace arrangements

The vanishing ideal I of a subspace arrangement V1union or logical sumV2union or logical sumcdots, three dots, centeredunion or logical sumVmsubset of or equal toV is an intersection I1∩I2∩cdots, three dots, centered∩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=I1I2cdots, three dots, centeredIm 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.