Hyperbolic polynomials and majorization
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2021-10-14Type of Work
16 pagesText
technical reports
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This item is likely protected under Title 17 of the U.S. Copyright Law. Unless on a Creative Commons license, for uses protected by Copyright Law, contact the copyright holder or the author.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