Smooth parametrizations for several cases of the Positivstellensatz (Q1365449)

The paper under review deals with a variation of the subject in a paper by \textit{C. N. Delzell}, \textit{L. González-Vega} and \textit{H. Lombardi} [in: Comput. Algebraic Geometry, Pap. Conf., Nice 1992, Prog. Math. 109, 61-75 (1993; Zbl 0804.14001)] and several Positivstellensatz cases. The three theorems proved concern: (a) Rational \(C^r\) parametrization of Hilbert's \(17^{\text{th}}\) problem; (b) Parametrizations for a variant of the homogeneous Hilbert's \(17^{\text{th}}\) problem; (c) Parametrization of the weak real homogeneous Nullstellensatz. In order to prove the results, a new and more general version for the homogeneous Nullstellensatz is proved.