Bit complexity for computing one point in each connected component of a smooth real algebraic set

We analyze the bit complexity of an algorithm for computation at least one point in each connected component a smooth real algebraic set. This work is continuation our analysis hypersurface case (On finding points components hypersurface, ISSAC'20). In this paper, we extend to more general cases. Let F=(f1,…,fp) Z[X1,…,Xn]p be sequence polynomials with V=V(F)⊂Cn and equidimensional variety 〈F〉⊂C[X1,…,Xn] radical ideal. To compute V∩Rn, starting by Safey El Din Schost (Polar varieties set, ISSAC'03). uses random changes variables that are proven generically ensure certain desirable geometric properties. The cost was given model; here, error probability, provide quantitative genericity statements. particular, led use Lagrange systems describe polar varieties, as they make it simpler rely on techniques such weak transversality effective Nullstellensatz.