@KConrad that works: people.math.gatech.edu/~mbaker/pdf/Coleman_GaloisNewton.pdf contains the proof in a very approachable way.

@KConrad: any available references to that work?

@YCor: thank you for pointing that out. Non-uniqueness is not an issue, just need one, or more, family of functions parametrized by $n$ with $A_n$ as the Galois Group.

@KConrad: the other question is more specific. I posted a generalized version here in the hopes that I may be pointed to some research on the topic. I will include a reference if I post again in a situation similar to this one.

@DavidESpeyer: I will update the question. Could you please provide a reference/proof/sketch-of-the-proof for that fact? Thank you.

@GerhardPaseman, looking at the vector representation and counting those with properties that place them on the boundaries is the only "reasonable" way I know of looking at it. A face of the hypercube has all point lying on some hyperplane at $x_i=0$ or $x_i=d$. Using higher level geometry, or fancier math, seems unnecessary. Just like in 3D $y=0$ means zx-plane, or $z=0$ means xy-plane, and so on.

I think we can easily see the first case is indeed 4: 3 triangles all with one corner at the origin and then the entire square. For $n=3$, that is the right count if you realize that polytopes need not be a triangle, you could choose 4 points including the origin and get a polypote with 4 sides where an edge would be part of the boundary, there are many such cases that only counting triangles leaves out..

Fixed. Thanks! (Also, the 15 character comment length minimum is silly.)