Advances in cylindrical algebraic decomposition

Cylindrical Algebraic Decomposition (CAD) was initially introduced to tackle the classic problem of quantifier elimination over real closed algebraic fields, however it has since seen many applications in its own right. Given a set of polynomials, multiple algorithms exist to produce a CAD such that over each cell the polynomials have constant sign. Inherently doubly exponential in the number of variables present, much work has been done to make CAD a practical tool through preconditioning, more efficient construction and truncated algorithms. I will give a brief history of CAD before covering work conducted by the University of Bath real geometry research group. Recently, we have shifted emphasis to try and produce a CAD for a given problem rather than the set of polynomials involved. A major step forward is research on Truth Table Invariant CADs (TTICADs) for which a set of Tarski clauses have invariant truth value over each cell. This research has also led to further investigation of how problems are formulated for input into various related algorithms. Alongside new research, key applications will be discussed. In particular, recent work on the use of CAD to verify identities involving multivalued functions over the complex numbers will be described. This work will be included in the forthcoming release of the computer algebra system Maple 17.