Computing real radicals and S-radicals of polynomial systems

Let f=(f1,…,fs) be a sequence of polynomials in Q[X1,…,Xn] of maximal degree D and V⊂Cn be the algebraic set defined by f and r be its dimension. The real radical 〈f〉re associated to f is the largest ideal which defines the real trace of V. When V is smooth, we show that 〈f〉re, has a finite set of generators with degrees bounded by deg⁡V. Moreover, we present a probabilistic algorithm of complexity (snDn)O(1) to compute the minimal primes of 〈f〉re. When V is not smooth, we give a probabilistic algorithm of complexity sO(1)(nD)O(nr2r) to compute rational parametrizations for all irreducible components of the real algebraic set V∩Rn.