Multiplicity computation of modules overk[x1,…,xn] and an application to weyl algebras

Abstract

Let A = k[x1,…,xn] be the polynomial algebra over a field kof characteristic 0Ian ideal of A, M = A/Iand αHP I the (affine) Hilbert polynomial of M. By further exploring the algorithmic procedure given in [CLO'] for deriving the existence of αHP I , we compute the leading coefficient of αHP I by looking at the leading monomials of a Grobner basis of Iwithout computing αHP I . Using this result and the filtered-graded transfer of Grobner 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 A n (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.