# Deduce Orientation of Point Configurations of a Polytope from the Face Lattice

Following the book of Juergen Richter-Gebert and this presentation article of Juergen Richter-Gebert and Guenther Ziegler, I am trying to compute the Realization Space of a polytope. I am considering a polytope P. For example the square given by the four points $A=(0,0), B=(1,0), C=(1,1), D=(0,1)$. Now I want to compute the realization space of this polytope. In oder to do so, I fix a basis for this polytope, consisting of $d+1$ vertices, where d is the dimension of the polytope, i.e. I fix three vertices, say $A,B,D$. Now let the coordinates of the fourth point be $C=(x,y)$. Then we homogenize the point configuration and we obtain the matrix $\left(\begin{array}{cccc}0&1&x&0\\0&0&y&1\\1&1&1&1\end{array}\right)$. According to my sources, the coordinates $x,y$ are restrited by certain $d+1\times d+1$ minors of this matrix, that have to be $>0$, $<0$ or $=0$. So far, this is fine, since:

1. $\left|\left(\begin{array}{ccc}0&1&x\\0&0&y\\1&1&1\end{array}\right)\right|=y$,
2. $\left|\left(\begin{array}{ccc}0&x&0\\0&y&1\\1&1&1\end{array}\right)\right|=x$,
3. $\left|\left(\begin{array}{ccc}1&x&0\\0&y&1\\1&1&1\end{array}\right)\right|=y+x-1$

Since we are given coordinates $C=(1,1)$, we can compute the determinants for these coordinates and then we know that the inequalities must be $y>0$, $x>0$, $y+x-1>0$.

The question is now: How can one deduce from the face lattice only, not using a given concrete point, whether the determinants have to be $>0$, $<0$ or $=0$?

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