And 3) The irreducibility is hard to model in this framework, but over an algebraically closed field there is of course the Nullstellensatz which can give infeasibility certificates.

Two comments: 1) This depends very much on the field you want your solutions from. Real root finding for instance is a branch of real algebraic geometry. 2) Gröbner bases can't help you since you can't use them to decide if a single polynomial is irreducible. You need factorization algorithms and then 1) comes into play.

Thank you! I've taken the liberty to edit your proof a little. As you say, it's not short, but quite elementary. Nice.

Can you give some more examples: How should that flatness measure be different from just height?