Computing real radicals and S-radicals of polynomial systems

Abstract

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.Let (g1,…,gp) in Q[X1,…,Xn] and S be the basic closed semi-algebraic set defined by g1≥0,…,gp≥0. The S-radical of 〈f〉, which is denoted by 〈f〉S, is the ideal associated to the Zariski closure of V∩S. We give a probabilistic algorithm to compute rational parametrizations of all irreducible components of that Zariski closure, hence encoding 〈f〉S. Assuming now that D is the maximum of the degrees of the fi's and the gi's, this algorithm runs in time 2p(s+p)O(1)(nD)O(rn2r). Experiments are performed to illustrate and show the efficiency of our approaches on computing real radicals.