integers $i$ and $j$ with $0< i < \frac d 2$ and $d-i  < j <d$. If $\partial h_\ell \leq \ell$ for some $\ell \leq d-j$ then $\sum_{k=0}^d h_k$ is even.

Let $v_1,v_2,\dots,v_d$ be distinct vertices of $SP$. Since $i<j$, in the same way as in the proof of Theorem \[2-3\], it is enough to prove that $$x_{v_d} x_{v_{d-1}}\cdots x_{v_{j+1}} \in J + (\Theta).$$

Let $u_k=x_{v_d} x_{v_{d-1}} \cdots x_{v_{d-k+1}}$ for $k=0,1,\dots,d-j$, where $u_0=1$. We regard each $u_k$ as an element in $R=A/J$. Recall that $\ell$ is an integer such that $\ell \leq d-j$ and $\partial h_\ell \leq \ell$. Since $\overline \Theta$ is an l.s.o.p. of $R/C=A/(L+J) \cong K[\partial P]$, $$\dim_K \big(R/(C+\overline \Theta R)\big)_\ell = \partial h_\ell \leq \ell.$$ Then the elements $$u_\ell, u_{\ell-1} \theta_d ,u_{\ell-2} \theta_d^2,\dots,u_0 \theta_d^\ell \in R_\ell$$ are $K$-linearly dependent in $R/(C+\overline \Theta R)$. Thus there are $\alpha_0,\alpha_1,\dots,\alpha_{\ell} \in K$ with $(\alpha_0,\alpha_1,\dots,\alpha_{\ell}) \ne 0$ such that $$\begin{aligned}
\alpha_0 u_\ell  + \alpha_1 u_{\ell-1} \theta_d + \cdots + \alpha_\ell u_0 \theta_d^\ell
\in 
C+\overline \Theta R.\end{aligned}$$ Let $s = \min\{ k : \alpha_k \ne 0\}$ and $h=(\sum_{k=s+1}^{\ell} \alpha_k u_{\ell-k} \theta_d^{k})/\theta_d^{s+1}$, where $h=0$ if $s=\ell$. Then $\theta_d^{s} (\alpha_{s} u_{\ell-s}+ \theta_d h ) \in (C+ \overline \Theta R)_\ell$ and $$\theta_d^{d-j-(\ell-s)} (\alpha_{s} u_{\ell-s}+ \theta_d 