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Systematically outputting sign vectors of restricted hyperplane arrangement

I have a way of constructing hyperplane arrangements, which involves taking the restriction of a larger arrangement to one of the hyperplanes in it. I want to output a list of the sign vectors of the regions of this arrangement, where the order of the coordinates in the sign vector corresponds to a specific order I want to impose on the hyperplanes. This has turned out surprisingly tricky, because of a couple of seemingly arbitrary decisions Sage's hyperplane arrangement package makes:

  • The hyperplane arrangement constructor seems to not preserve the order of the hyperplanes fed into it, so if I just run the built-in sign_vector method, the coordinates of the resulting vectors may be rearranged from how I want them.
  • On the other hand, I'm not sure how to go about manually determining the sign vector, because I'm not sure how Sage decides to coordinatize the hyperplanes in the restricted arrangement. So while I have equations for the hyperplanes in the original ambient space, I'm not sure how I could automatically turn them into equations in the hyperplane I'm restricting to.

Does anyone have advice on how to systematically resolve either of these arbitrary decisions, or how to find a different workaround?