  Polynomial Systems, or at least their algorithms, have the reputation of
being doubly-exponential in the number of variables [Mayr and Mayer, 1982],
[Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that
number of zeros of a zero-dimensional system is singly-exponential in the
number of variables. How should this contradiction be reconciled?
  We first note that [Mayr and Ritscher, 2013] shows that the doubly
exponential nature of Gr\"{o}bner bases is with respect to the dimension of the
ideal, not the number of variables. This inspires us to consider what can be
done for Cylindrical Algebraic Decomposition which produces a
doubly-exponential number of polynomials of doubly-exponential degree.
  We review work from ISSAC 2015 which showed the number of polynomials could
be restricted to doubly-exponential in the (complex) dimension using McCallum's
theory of reduced projection in the presence of equational constraints. We then
discuss preliminary results showing the same for the degree of those
polynomials. The results are under primitivity assumptions whose importance we
illustrate.
