The Geometry of Hyperbolic Polynomials and an Application

Abstract

This thesis aims to expand on the little known but rich mathematics of hyperbolic polynomials. Our two main results lie in rather different fields of mathematics. First we sit in algebraic geometry, looking at the space of hyperbolic polynomials themselves. We prove a result giving a class of operators on the space of polynomials preserving hyperbolicity. Our second result moves into differential geometry where we use G˚arding’s inequality for hyperbolic polynomials to prove a rigidity theorem for spacelike hypersurfaces in de Sitter space. To fix notation and give a feel for what hyperbolic polynomials are, we begin by giving an exposition of the theory introduced by G˚arding in 1959.