  These are lecture notes for a course I gave in mid-1990s for MSc students at
the University of Bath. It presents an algorithm with singly exponential
complexity for the existential theory of the reals, in the spirit of J.
Renegar. The aim was to convey the main underlying ideas, so many of the proofs
and finer details of algorithms are either missing or just sketched. I changed
nothing in the original notes except adding references, bibliography, and
correcting obvious typos.
