Abstract
This paper gives a description of some aspects of complexity in real algebraic geometry: explicit bounds for the topology of real algebraic and semi-algebraic sets are given in term of the degree of the equations and inequations (results by Milnor-Thom), and then in term of the additive complexity of the equations and inequations (i.e. in term of the minimal number of addition-substractions needed to evaluate the equations) (results by Grigorlev-Risler). Those results are then used to give lower bounds for the complexity of various algorithms (results of Ben-Or). The work of Collins and its school about C.A.D. of semi-algebraic sets is mentioned, and the paper ends with a description of a class of real analytic sets to which it is possible to generalise some of the previous methods (results by Hovansky).
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