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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In this paper we prove new bounds on the sum of the Betti numbers of closed semi-algebraic sets and also give the first single exponential time algorithm for computing the Euler characteristic of arbitrary closed semi-algebraic sets.Given a closed semi-algebraic set S ⊂ R k defined as the intersection of a real variety, Q = 0, deg(Q) ≤ d, whose real dimension is k , with a set defined by a quantifierfree Boolean formula with no negations with atoms of the form P i = 0, P i ≥ 0, P i ≤ 0, deg(P i ) ≤ d, 1 ≤ i ≤ s, we prove that the sum of the Betti numbers of S is bounded by s k (O(d)) k . This result generalizes the Oleinik-Petrovsky-Thom-Milnor bound in two directions. Firstly, our bound applies to arbitrary unions of basic closed semi-algebraic sets, not just for basic semi-algebraic sets. Secondly, the combinatorial part (the part depending on s) in our bound, depends on the dimension of the variety rather than that of the ambient space. It also generalizes the result in [4] where a similar bound is proven for the number of connected components. We also prove that the sum of the Betti numbers of S is bounded by s k 2 O(k 2 m 4 ) in case the total number of monomials occurring in the polynomials in P ∪ {Q} is m. Using the tools developed for the above results, as well as some additional techniques, we give the first single exponential time algorithm for computing the Euler characteristic of arbitrary closed semi-algebraic sets.

Let R be a real closed field, P, Q ⊂ R[X 1 , . . . , X k ] finite subsets of polynomials, with the degrees of the polynomials in P (resp., Q) bounded by d (resp., d 0 ). Let V ⊂ R k be the real algebraic variety defined by the polynomials in Q and suppose that the real dimension of V is bounded by k . We prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of the family P on V is bounded by k j =0