Hyperbolic polynomials and majorization

Abstract

On a finite dimensional real vector space V, we consider a real homogeneous polynomial p of degree n that is hyperbolic relative to a vector e ∈ V. This means that p(e) 6= 0 and for any (fixed) x ∈ V, the roots of the one-variable polynomial t 7→ p(te − x) are all real. Let λ(x) denote the vector in Rn whose entries are these real roots written in the decreasing order. Relative to the map λ : V → Rn, we introduce and study automorphisms, majorization, and doubly stochastic transformations