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.

Grünbaum, Barnette, and Reay in 1974 completed the characterization of the pairs (f i , f j ) of face numbers of 4-dimensional polytopes.Here we obtain a complete characterization of the pairs of flag numbers (f 0 , f 03 ) for 4-polytopes. Furthermore, we describe the pairs of face numbers (f 0 , f d−1 ) for d-polytopes; this description is complete for even d ≥ 6 except for finitely many exceptional pairs that are "small" in a well-defined sense, while for odd d we show that there are also "large" exceptional pairs.Our proofs rely on the insight that "small" pairs need to be defined and to be treated separately; in the 4-dimensional case, these may be characterized with the help of the characterizations of the 4-polytopes with at most 8 vertices by Altshuler and Steinberg (1984).

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