Different notions of triangular sets are presented. The relationship between these notions are studied. The main result is that four different existing notions of good triangular sets are equivalent.

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 ⊂ R[y1, . . . , yn] 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. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.

We present lifting techniques for triangular decompositions of zero-dimensional varieties, that extend the range of the previous methods. We discuss complexity aspects, and report on a preliminary implementation. Our theoretical results are comforted by these experiments.

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