  Cylindrical Algebraic Decomposition (CAD) is an important tool within
computational real algebraic geometry, capable of solving many problems to do
with polynomial systems over the reals, but known to have worst-case
computational complexity doubly exponential in the number of variables. It has
long been studied by the Symbolic Computation community and is implemented in a
variety of computer algebra systems, however, it has also found recent interest
in the Satisfiability Checking community for use with SMT-solvers. The SCSC
Project seeks to build bridges between these communities.
  The present report describes progress made during a Research Internship in
Summer 2017 funded by the EU H2020 SCSC CSA. We describe a proof of concept
implementation of an Incremental CAD algorithm in Maple, where CADs are built
and refined incrementally by polynomial constraint, in contrast to the usual
approach of a single computation from a single input. This advance would make
CAD of use to SMT-solvers who search for solutions by constantly reformulating
logical formula and querying solvers like CAD for whether a logical solution is
admissible. We describe experiments for the proof of concept, which clearly
display the computational advantages when compared to iterated re-computation.
In addition, the project implemented this work under the recently verified
Lazard projection scheme (with corresponding Lazard evaluation). That is the
minimal complete CAD method in theory, and this is the first documented
implementation.
