We give a new algorithm, with three versions, for computing the number of real roots of a system of n polynomial equations in n unknowns. The rst version is of Monte Carlo type and, neglecting logarithmic factors, runs in time quadratic in the average number of complex roots of a closely related system. The other two versions run nearly as fast and progressively remove a measure zero locus of failures present in the rst version. Via a slight simpli cation of our algorithm, we can also count

more »

... lex roots, with or without multiplicity, within the same complexity bounds. We also derive an even faster algorithm for the special case n=2, which may be of independent interest. Remark 1 For technical reasons, we will only consider roots with all coordinates nonzero. This restriction will be lifted in forthcoming work of the author, and most of the theory necessary for this extension has already appeared in Roj97c,Roj98a]. We will also assume that all our input polynomial systems have only nitely many complex roots with all coordinates nonzero. This mild restriction can also be lifted, but the details will be covered in a later paper. In any case, a feature of our algorithm is that we do allow in nitely many roots at in nity. 1 Our Main Theorem, along with an independent approach in MP98], represent the rst complexity bounds for real root counting which are intrinsic in the sense that they depend mainly on the geometric properties of the underlying ? Extended abstract