Example

Let $A$ be a ring of formal series $k[[X_1, ..., X_n]]$ over a field $k$; let $F_1, ..., F_s$ be elements of A, a the ideal which they generate, and $B = A/a$. Let $P_1, ..., P_s \in k[X_1, ..., X_n]$ be the initial forms of the series $F_1, ..., F_s$ and $\delta_1, ..., \delta_s$ their respective degrees. If the sequence $F_1, ..., F_s$ is secant in A, one has $e(B) \geq \delta_1 ... \delta_s$; if the sequence $P_1, ..., P_s$ is completely secant in the ring $k[X_1, ..., X_n]$, one has $e(B) = \delta_1 ... \delta_s$.

Consider for example the ring $B = k[[X, Y]]/a$, where a is generated by $X^2 + Y^3$ and $X^2 + Y^4$; the preceding inequality gives $e(B) \geq 4$; observing that a is generated by the elements $X^2 + Y^3$ and $Y^4 - Y^3$, for which the sequence of initial forms is completely secant, one obtains $e(B) = 6$.