poset $SP$ by $$SP= P \cup C(\partial P)/\! \sim$$ where $\sim$ is the equivalence relation defined by $\sigma \sim (\sigma,1)$ for all $\sigma \in \partial P$. In other words, $SP$ is a simplicial cell sphere such that $\Gamma(SP)$ is obtained from $\Gamma(P)$ by coning off its boundary.

We need the next result due to Masuda [@Ma Section 5] (see also [@MR] for a simplified proof).

\[2-2\] Let $P$ be a simplicial cell $(d-1)$-sphere, $K[P]=A/I_P$ the face ring of $P$ and $\Theta=\theta_1,\dots,\theta_d \in A_1$ an l.s.o.p. of $K[P]$. If $x_{v_1} x_{v_2} \cdots x_{v_d}$ is zero in $A/(I_P+(\Theta))$ for any sequence of $d$ distinct vertices $v_1,v_2,\dots,v_d$ of $P$ then the number of the facets of $P$ is even.

In the rest of this section, we fix the following notation: Let $P$ be a simplicial cell $(d-1)$-ball and $${\mathbf{h}}=h(P)=(h_0,h_1,\dots,h_d).$$ Let $Q=C(\partial P)$ and $SP=P \cup C(\partial P)/\! \sim$ be simplicial posets defined as above. Also, let $A=K[x_\sigma: \sigma \in SP \setminus\{\hat 0\}]$, $$J=I_{SP}+(x_\sigma: \sigma \in SP \setminus P)=I_{SP}+(x_\sigma: \sigma \in Q \setminus \partial P),$$ $$L= I_{SP}+(x_\sigma: \sigma \in SP \setminus Q)
=I_{SP}+(x_\sigma: \sigma \in P \setminus \partial P),$$ and let $\Theta=\theta_1,\theta_2,\dots,\theta_d \in A_1$ be a sequence of general linear forms. Then we have 