In F. Beukers' Notes on A-Hypergeometric functions on p. 17, the Appell hypergeometric PDEs are derived from the GKZ-Equations \begin{align*} \partial_{1}\partial_{2}\Phi-\partial_{4}\partial_{5}\Phi & =0,\partial_{1}\partial_{3}\Phi-\partial_{4}\partial_{6}\Phi=0 \end{align*} and \begin{align*} (v_{1}\partial_{1}+v_{5}\partial_{5}+v_{6}\partial_{6}+a)\Phi & =0\\ (v_{2}\partial_{2}+v_{5}\partial_{5}+b)\Phi & =0\\ (v_{3}\partial_{3}+v_{6}\partial_{6}+b')\Phi & =0\\ (v_{4}\partial_{4}-v_{5}\partial_{5}-v_{6}\partial_{6}+1-c)\Phi & =0. \end{align*} The passage from the mentioned text I do not understand much is the one below these equations:
''Let $\mathcal{Z}$ be the left ideal in $K[\partial_{1},...,\partial_{6}]$ generated by the operators $Z_{1},...,Z_{4}$ (here $K=C(v_{1},...,v_{6})$). Consider for each box operator $\square$ the intersection of the class $\square(\mod\mathcal{Z})$ with $K[\partial_{5},\partial_{6}]$ and set $v_{1}=\cdot\cdot\cdot=v_{4}=1,v_{5}=x,v_{6}=y$.
We obtain the classical differential equations \begin{align*} x(1-x)F_{xx}+y(1-x)F_{xy}+(c-(a+b+1)x)F_{x}-byF_{y}-abF & =0\\ y(1-y)F_{yy}+x(1-y)F_{xy}+(c-(a+b'+1)y)F_{y}-b'xF_{x}-ab'F & =0. \end{align*} ''
Here, $K$ is a differential field with commuting derivations $\partial_{i}=\frac{\partial}{\partial v_{i}}$ for $i=1,...,N$, the box operator
\begin{align*} \square_{l}\Phi: & =\prod_{l_{i}>0}\partial_{i}^{l_{i}}\Phi-\prod_{l_{i}<0}\partial_{i}^{|l_{i}|}\Phi=0 \end{align*} and \begin{align*} Z_{i}\Phi: & =(a_{i1}v_{1}\partial_{1}+a_{i2}v_{2}\partial_{2}+\text{·}\text{·}\text{·}+a_{iN}v_{N}\partial_{N}-\alpha_{i})\Phi=0,\quad i=1,2,...,r\ . \end{align*}
Can someone explain the very basic steps needed to derive Appell's PDEs from these GKZ-equations? I am not yet familiar with abstract algebra (nor with differential algebra).
I am physicist trying to understand whether the GKZ-functions help me to solve linear second-order PDEs with polynomial coefficients more general than the coefficients of Appell's PDEs. If this is the case, I will invest more time to study these functions and the topic of differential algebra.