  In [4] Sturmfels linked the Hilbert Nullstellensatz to Gr\"obner bases
through final polynomials. In (loc. cit.) it was claimed that final polynomials
always appear in a lexicographic Gr\"obner basis of a certain ideal. In this
paper, we give a counterexample to this claim. We also show how the
introduction of an extra variable restores the claim in a deformed setup, which
we call extended final polynomials.
