Semi-algebraic sets of f-vectors

Polytope theory has produced a great number of remarkably simple and complete characterization results for face-number sets or f -vector sets of classes of polytopes. We observe that in most cases these sets can be described as the intersection of a semi-algebraic set with an integer lattice. Such semi-algebraic sets of lattice points have not received much attention, which is surprising in view of a close connection to Hilbert's Tenth problem, which deals with their projections. We develop proof techniques in order to show that, despite the observations above, some f -vector sets are not semi-algebraic sets of lattice points. This is then proved for the set of all pairs ( f 1 , f 2 ) of 4-dimensional polytopes, the set of all f -vectors of simplicial d -polytopes for d ≥ 6, and the set of all f -vectors of general d -polytopes for d ≥ 6. For the f -vector set of all 4-polytopes this remains open.