Cylindrical algebraic decomposition using local projections

We present an algorithm which computes a cylindrical algebraic decomposition of a semialgebraic set using projection sets computed for each cell separately. Such local projection sets can be significantly smaller than the global projection set used by the Cylindrical Algebraic Decomposition (CAD) algorithm. This leads to reduction in the number of cells the algorithm needs to construct. A restricted version of the algorithm was introduced in Strzeboński (2014). The full version presented here can be applied to quantified formulas and makes use of equational constraints. We give an empirical comparison of our algorithm and the classical CAD algorithm.