\big)_k=h_k(Q)=h_k(\partial P)=\partial h_k\end{aligned}$$ for $k =0,1,\dots,d-1$. Thus, $\partial h_i=0$ implies $x_{v_{1}}x_{v_2} \cdots x_{v_i} \in L+(\Theta).$ Similarly, since $\dim_K (A/(J+(\Theta)))_j=h_j$, $h_j=0$ implies $x_{v_d} x_{v_{d-1}} \cdots x_{v_{d-j+1}} \in J+(\Theta)$.

Now we prove (5) and (6).

\[2-3\] If $\partial h_i=0$ and $h_j=0$ for some positive integers $i$ and $j$ with $i+j \leq d$ then $\sum_{k=0}^d h_k$ is even.

Let $v_1,v_2,\dots,v_d$ be distinct vertices of $SP$. By Lemmas \[2-2\] and \[facets\], it is enough to prove that the monomial $x_{v_1}x_{v_2}\cdots x_{v_d}$ is zero in $A/(I_{SP} +(\Theta))$.

Since $i+j \leq d$, Lemma \[int1\] shows $$x_{v_1} x_{v_2} \cdots x_{v_d} \in
LJ+(\Theta) \subset
 I_{SP} +(\Theta)+(x_\sigma x_\tau :
\sigma \in P \setminus \partial P,\
\tau \in Q \setminus \partial P).$$ However, for all $\sigma \in P \setminus \partial P$ and $\tau \in Q \setminus \partial P$, we have $x_\sigma x_\tau \in I_{SP}$ since $\sigma$ and $\tau$ have no common upper bounds. Thus $x_{v_1} x_{v_2} \cdots x_{v_d} \in I_{SP}+(\Theta)$.

\[necessary7\] Suppose $\partial h_n = 0$ for some $1 \leq n < \frac d 2$. If $(h_\ell + h_{\ell-1} + \cdots + h_{\ell-n+1}) -\partial h_\ell < n$ for some $ n \leq \ell \leq d-n$ then $\sum_{k=0}^d h_k$ is even.

Let $v_1,v_2,\dots,v_d$ be distinct vertices of $SP$. Then $x_{v_1} x_{v_2} 