Marc, the first principal minor in the Jacobian of $F$ is just the $x$-derivative of $f(x,y)$, and I don't believe that this quantity is non-vanishing. If you evaluate the derivative as a function of $y$ at $x=0$ for instance, it vanishes whenever $\tan(y) = \frac{1}{2y}$.

Pablo, thanks. I have modified the question to indicate that the sets generate, rather than exhaust, the sigma algebra. In any case, it seems as though Michael understood what I meant and has settled the second question.

Alexandre, this question is only about getting from a) to d). There are "literally hundreds of examples" because you have over-estimated the scope of the question. In particular, getting to stage c) is no guarantee whatsoever that stage d) will actually be reached and without that you are including all possible existence results for which no constructive version is known, rather than examples where the constructive version is known and came after many years. That being said, the K\'obe constant example sounds interesting and relevant.

@Nate: yes. Is there another way to interpret the question?

Claudio, I believe this has already been covered in Lee Mosher's answer.

Dear gruff, thank you for this nice answer. I have been going through Eisenbud in light of comments to my original question, but it is not yet clear what one "does" with this uniqueness of minimal resolutions. What is it good for? It is my understanding that any free resolution yields the hilbert function, and so minimality is completely unnecessary in this context. Does the minimality provide, for instance, any algorithmic benefits (as in computing Grobner bases via Buchberger)?

Quid: thank you for this information. I will run tests to see if the random simplicial complexes have similar statistics and update when I have results.

Jordan: I did toy with that idea but had no clue what the $c$ in your $c/p$ should be. I'm a huge fan of Matt's work, and would love to be able to apply it here.

n the context of general cell complexes over a coefficient ring R, one should note that discrete Morse theory becomes severely limited: for the simple homotopy equivalence to hold, the discrete Vector field is only allowed to pair adjacent cells $\sigma < \tau$ if the degree of the attaching map from the boundary of $\tau$ onto $\sigma$ is a unit as well as a central element of $R$. This is made clear in the work of Welker etc. here in a purely algebraic setting: www.maths.ed.ac.uk/~aar/papers/jollwelk.pdf

Dear Zack, how are you sure that the new cycles created by the introduction of 43 are not related via the boundary of a 2-chain to each other or the old cycles? That is, why are all the 8 cycles you mention in K[43] necessarily non-homologous? In any case, I have modified the question to indicate that not only the codimension-1 facets, but all faces and sub-faces must be inserted into the simplicial