Discriminants and nonnegative polynomials (Q655567)

Let \(K\subset\mathbb{R}^n\) be a semialgebraic set, and \(P_d(K)\) the cone of multivariate polynomials with real coefficients and multidegree bounded by \(d\) that are nonnegative on \(K\). The paper under review studies the geometry of the boundary of \(P_d(K)\). It is shown that the cone is also a semialgebraic set, and that \(\partial P_d(K)\) is a hypersurface defined by a polynomial equation. If \(K=\mathbb{R}^n\) and \(d>2\) is even, then \(\partial P_d(K)\) is contained in the irreducible discriminantal hypersurface associated to a generic multivariate polynomial of multidegree \(d\). This discriminant was defined and studied in [\textit{I. M. Gelfand, M. M. Kapranov} and \textit{A. V. Zelevinsky}, Discriminants, resultants, and multidimensional determinants. Boston, MA: Birkhäuser (1994; Zbl 0827.14036); Reprint of the 1994 edition. Boston, MA: Birkhäuser (2008; Zbl 1138.14001)]. It is also shown that if \(K\) is a real algebraic variety, then its border lies on another discriminantal hypersurface related to the defining polynomials of \(K\). In general, the author shows that \(\partial P_d(K)\) lies on a union of several discriminantal hypersurfaces. It is also shown that typically these cones do not have a barrier of type \(-\log\varphi(f)\) if \(\varphi(f)\) is required to be a polynomial, but such a barrier exists if \(\varphi(f)\) is allowed to be semialgebraic. The paper concludes with illustrations and examples.