Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set F of polynomials with N variables with real coffecients, we apply comprehensive triangular decomposition in order to obtain an F-invariant cylindrical decomposition of the n-dimensional complex space, from which we extract an F-invariant cylindrical algebraic decomposition of the n-dimensional real space.

This new approach for constructing cylindrical algebraic decompositions has been implemented in the RegularChains library in Maple. We shall demonstrate its usage together with the other tools of this library for solving polynomial systems arising in real life problems.