We define multiplicity of a module $M$ of dimension $d>0$ as $$e(M) := \operatorname{lc} (P_M) (d-1)!,$$ where $P_M$ denotes the Hilbert polynomial of $M$ and $\operatorname{lc}(P_M)$ its leading coefficient. Equivalently, we have $e(M) = Q_M(1)$, where $HP_M (z) = \frac{Q_M(z)}{(1-z)^d}$ and $HP_M$ is the Hilbert-Poincaré series of $M$.
I can prove that if $I = (f_1, \dots, f_r)$, with $\deg(f_i)=d_i$ and $f_1, \dots, f_r$ is an $M$-regular sequence, then $$e(M/IM) = d_1 \cdots d_r \cdot e(M)$$
Consider now $R= \mathbb{k}[x_1, \dots, x_n]$, $I = (f_1, \dots, f_r)$, with $\deg(f_i)=d_i \geq 2$ and $f_i$ homogeneous. Does it holds the reverse implication, i.e. $$ e(R/I) = d_1 \cdots d_r \quad \Rightarrow \quad f_1,\dots,f_r \quad \text{is a $R$-regular sequence?}$$