Abstract
A presentation of the quantifier elimination and other problems connected to it, such as the piano mover's problem, is followed by the state of the art on the question. Afterwards are explained Schwartz', Sharir's and Collins' algorithm, based on the cylindrical algebraic decomposition, and two applications to bring out its possibilities and limitations. The tracks and views of development in this research domain, which is a useful tool for algebraic geometry, are lastly indicated. Its limitation, due to a very high cost, is an actual obstacle, which is to be removed, for its use for many classes of concrete geometric and algebraic problems.
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