The present edition is a substantially revised and expanded English version of the book "Geometrie algebrique reelle", originally published in French as Ergebnisse der Mathematik und ihrer

We survey both old and new developments in the theory of algorithms in real algebraic geometry -starting from effective quantifier elimination in the first order theory of reals due to Tarski and Seidenberg, to more recent algorithms for computing topological invariants of semi-algebraic sets. We emphasize throughout the complexity aspects of these algorithms and also discuss the computational hardness of the underlying problems. We also describe some recent results linking the computational hardness of decision problems in the first order theory of the reals, with that of computing certain topological invariants of semi-algebraic sets. Even though we mostly concentrate on exact algorithms, we also discuss some numerical approaches involving semi-definite programming that have gained popularity in recent times. Contents

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