There is an active area of research in effective and computational algebraic geometry devoted to these sorts of questions (and in many other fields than algebraic geometry also), but I don’t know the latest results. Certainly, a naive quantification of the textbook qualitative arguments here would lead to various unnecessary exponential losses, but I don’t know if by being careful enough one can eradicate all such exponential factors. (It may be that one has to carefully word the result in order to get, say, polynomial bounds, as many concepts that are equivalent at the exponential bound level or above may become inequivalent at the polynomial bound level.)

]]>