  Let $M$ be a finitely generated module on a local ring $R$ and $\F:
M_0\subset M_1\subset...\subset M_t=M$ a filtration of submodules of $M$ such
that $ d_o<d_1< ... <d_t=d$, where $d_i=\dim M_i$. This paper is concerned with
a non-negative integer $p_\mathcal F(M)$ which is defined as the least degree
of all polynomials in $n_1, ..., n_d$ bounding above the function
$$\ell(M/(x_1^{n_1}, ..., x_d^{n_d})M)-\sum_{i=0}^tn_1...n_{d_i}e(x_1,...,
x_{d_i};M_i).$$ We prove that $p_\mathcal F(M)$ is independent of the choices
of good systems of parameters $\underline x=x_1, ..., x_d$. When $\F$ is the
dimension filtration of $M$ we also present some relations between $p_\F(M)$
and the polynomial type of each $M_i/M_{i-1}$ and the dimension of the
non-sequentially Cohen-Macaulay locus of $M$.
