Zero estimates with moving targets

Zero estimates have a classical history in diophantine approximation and transcendence, as well as more recent applications to counting rational points on analytic sets. We give some examples showing that the sharpest conceivable results can be false, and in some cases the natural guess has even to be doubled. A by-product is a "non-degenerate" case of the (unproved) "Zilber Nullstellensatz" in connexion with "Strong Exponential Closure".