Discriminants and nonnegative polynomials

Abstract: For a semialgebraic set in , let be the cone of polynomials in of degrees at most that are nonnegative on . This paper studies the geometry of its boundary . We show that when and is even, its boundary lies on the irreducible hypersurface defined by the discriminant of . We show that when is a real algebraic variety, lies on the hypersurface defined by the discriminant of . We show that when is a general semialgebraic set, lies on a union of hypersurfaces defined by the discriminantal equations. Explicit formulae for the degrees of these hypersurfaces and discriminants are given. We also prove that typically does not have a barrier of type when is required to be a polynomial, but such a barrier exists if is allowed to be semialgebraic. Some illustrating examples are shown. [Copyright &y& Elsevier]