  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 $d\ge6$, and the set
of all f-vectors of general $d$-polytopes for $d\ge6$. For the f-vector set of
all 4-polytopes this remains open.
