Computing The Real Radicals Of Higher-dimensional Polynomial Ideals

The real radicals of polynomial ideals play important roles in real algebraic geometry.These radicals are closely related to the well-known real Nullstellensatz.So the computation of real radicals becomes an active research area in computational real algebraic geometry.Set a proper ideal I in polynomial ring K[x1,x2,...,xn],and we will compute the real radical(?)of I in this paper.At first,we briefly introduce the computation of real radicals of zero-dimensional ideals from three cases:the univariate case,the single polynomial case and multivariate case.Then,we will describe an algorithm to compute the real radical of higher-dimensional ideal in detail.To compute the real radical of higher-dimensional ideal I in polynomial ring K[x1,x2,...,Xn],at first,we compute a set{x1,x2,...,xs} that is maximally independent modul I.And then we will extend the higher-dimensional ideal I in ring K[x1,x2,...,xn]into zero-dimensional ideal Ie in ring K(x1,x2,...,xs)[xs+1,xs+2,...,xn]via the canonial ring homomorphism:?:K[x1,x2,...,xn]?K(x1,x2,...,xs)[xs+1,xs+2,...,xn](1?s?n-1).Then compute the real radical of zero-dimensional polynomial ideal Ie in K(x1,x2,...,xs)[xs+1,xs+2,...,xn].At last,we can contract it back in K[x1,x2,...,xn],then the real radical of higher-dimensional ideal I is computed,and the algorithm will be described by an example.