Multiplicity computation of modules over k[x1, ..., xn] and an application to Weyl algebras

Abstract

Let A = k[x1,...,xn] be the polynomial algebra over a field k of characteristic 0, I an ideal of A, M = A/I and aHPI the (affine) Hilbert polynomial of M. By further exploring the algorithmic procedure given in [CLO'] for deriving the existence of aHPI, we compute the leading coefficient of aHPI by looking at the leading monomials of a Gröbner basis of I without computing aHPI. Using this result and the filtered-graded transfer of Gröbner basis obtained in [LW] for (noncommutative) solvable polynomial algebras (in the sense of [K-RW]), we are able to compute the multiplicity of a cyclic module over the Weyl algebra An (k) without computing the Hilbert polynomial of that module, and consequently to give a quite easy algorithmic characterization of the "smallest" modules over Weyl algebras. Using the same methods as before, we also prove that the tensor product of two cyclic modules over the Weyl algebras has the multiplicity which is equal to the product of the multiplicities of both modules. The last result enables us to construct examples of "smallest" irreducible modules over Weyl algebras.