Throughout this paper, let M be a finitely generated module over a Noether local ring (A, [mfr ]) with dim M = d. Let x = (x1, …, xd) be a system of parameters of M and n = (n1, …, nd) ∈ ℕd a d-tuple of positive integers. This paper is concerned with the following two points of view.

First, it is well-known that, the difference between lengths and multiplicities

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considered as a function in n, gives a lot of information on the structure of M. This function in general is not a polynomial in n for all n1, …, nd large enough (n [Gt ] 0 for short). But, it was shown in [C3] that the least degree of all polynomials in n bounding above IM (n, x) is independent of the choice of x. This numerical invariant is denoted by p(M) and called the polynomial type of the module M. By [C2] and [C3] this polynomial type does not change by the [mfr ]-adic completion Mˆ of M and p(M) is just equal to the dimension of the non-Cohen–Macaulay locus of Mˆ (see [C2, C3, C4, CM] for more details). Therefore a module M is Cohen–Macaulay or generalized Cohen–Macaulay if and only if p(M) = − ∞ or p(M) [les ] 0, respectively, where we set by − ∞ the degree of the zero-polynomial. However, one knows little about the structure of M when p(M) > 0.

Second, following Sharp and Hamieh ([SH]), we consider the difference

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as a function in n, where

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is the cyclic submodule of the module of generalized fractions U(M)−d−1d+1M defined in [SZ1].