Let $S=K[x_1,\cdots,x_s]$ be a polynomial ring over a characteristic zero field $K$. For a partition $\{1, \cdots, s\} = I_1 \sqcup \cdots \sqcup I_r$ of the variable index, let $\mathbb{x}_i^{\mathbb{a}_i} := \prod_{k \in I_i} x_k^{a_k}$ be the monomial of variables for $I_i$ with multi-degree $\mathbb{a}_i = (a_k)_{k \in I_i}$ for $1 \leq i \leq s$. For each $1 \leq i \leq s$, the total degree of $\mathbb{a}_i$ is defined to be $|\mathbb{a}_i| := \sum_{k \in I_i} a_k$. We equip $S$ with a filtration valued in $\mathbb{N}^r$ as for $\forall \lambda \in \mathbb{N}^r$, $$F_\lambda S := \bigoplus_{\mu \leq \lambda} \bigoplus_{\mathbb{a}_i | |\mathbb{a}_i| = \mu_i} K \cdot \prod_i^r \mathbb{x}_i^{\mathbb{a}_i} = \bigoplus_{\mathbb{a}_i | |\mathbb{a}_i| \leq \lambda_i} K \cdot \prod_i^r \mathbb{x}_i^{\mathbb{a}_i}.$$ Let $M$ be a finitely generated module over $S$ with a set of generators $\{m_1,\cdots,m_l\}$. We define an increasing filtration on $M$ by setting $F_\lambda M := \sum\limits_{i=1}^l F_\lambda S \cdot m_i$.
My question is: what do we know about $\mathrm{dim}_K F_\lambda M$? Does it grows like a polynomial with homogenous leading terms of degree $\mathrm{dim} ~ M$? Or do we know a upper bound by such a polynomial?
