Ideals, bifiltered modules and bivariate Hilbert polynomials

Let R be a ring of polynomials in m+n variables over a field K and let I be an ideal in R. Furthermore, let be the natural bifiltration of the ring R and let be the corresponding natural bifiltration of the R-module M=R/I associated with the given set of generators introduced by Levin. The author shows an algorithm for constructing a characteristic set G={g1,…,gs} of I with respect to a special type of reduction introduced by Levin, that allows one to find the Hilbert polynomial in two variables of the bifiltered and bigraded R-module R/I. This algorithm can be easily extended to the case of bifiltered R-submodules of free R-modules of finite over R.