I have a $d$-dimensional finite abstract simplicial complex (ASC); that is I have a sequence of incidence matrices that define the complex. I know that any such ASC can be embedded in ${\mathbb R}^N$ for $N$ approximately the number of vertices. I'm interested in embeddings in ${\mathbb R}^d$ (or ${\mathbb R}^{d+1,d}$). Is there an algorithm to find such an embedding or determine if it doesn't exist?
Following the suggestion in the comments I'm adding a concrete example of what I mean by incidence matrices. The complex here corresponds to the 3d associahedron (of type A) :
C[1] is 1x9 matrix
[[1,1,1,1,1,1,1,1,1]]
C[2] is 9x21 matrix
[[1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[1,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,1,0,0,0,1,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,0,0,0,0,0],
[0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0],
[0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,1,0,0,0],
[0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,1,1,0],
[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,1]]
C[3] is 21x14 matrix
[[1,1,0,0,0,0,0,0,0,0,0,0,0,0],
[1,0,1,0,0,0,0,0,0,0,0,0,0,0],
[0,0,1,1,0,0,0,0,0,0,0,0,0,0],
[0,1,0,0,1,0,0,0,0,0,0,0,0,0],
[0,0,0,1,1,0,0,0,0,0,0,0,0,0],
[1,0,0,0,0,1,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,1,0,0,0,0,0,0,0],
[0,1,0,0,0,0,1,0,0,0,0,0,0,0],
[0,0,0,0,0,1,0,1,0,0,0,0,0,0],
[0,0,1,0,0,0,0,0,1,0,0,0,0,0],
[0,0,0,0,0,0,0,1,1,0,0,0,0,0],
[0,0,0,0,0,0,1,0,0,1,0,0,0,0],
[0,0,0,0,0,0,0,1,0,0,1,0,0,0],
[0,0,0,0,0,0,0,0,0,1,1,0,0,0],
[0,0,0,0,0,0,0,0,1,0,0,1,0,0],
[0,0,0,1,0,0,0,0,0,0,0,1,0,0],
[0,0,0,0,1,0,0,0,0,0,0,0,1,0],
[0,0,0,0,0,0,0,0,0,1,0,0,1,0],
[0,0,0,0,0,0,0,0,0,0,0,1,0,1],
[0,0,0,0,0,0,0,0,0,0,1,0,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,1,1]]
C[4] is 14x1 matrix
[[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1]]