Hilbert series of subspace arrangements

The vanishing ideal II of a subspace arrangement V1∪V2∪⋯∪Vm⊆VV1∪V2∪⋯∪Vm⊆V is an intersection I1∩I2∩⋯∩ImI1∩I2∩⋯∩Im of linear ideals. We give a formula for the Hilbert polynomial of II if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J=I1I2⋯ImJ=I1I2⋯Im without any assumptions about the subspace arrangement. It turns out that the Hilbert series of JJ is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of JJ 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.