Gröbner bases and final polynomials

In [4] Sturmfels linked the Hilbert Nullstellensatz to Gröbner bases through final polynomials. In (loc. cit.) it was claimed that final polynomials always appear in a lexicographic Gröbner 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.