Very hyperbolic polynomials in one variable

Abstract

A real polynomial in one real variable is called hyperbolic if it has only real roots. The polynomial f is called a primitive of order ν of the polynomial g if f^(ν) = g. A hyperbolic polynomial is called very hyperbolic if it has hyperbolic primitives of all orders. In the paper we prove some geometric properties of the set D of values of the parameters a_i for which the polynomial xⁿ + a₁xⁿ⁻¹ + … + a_n is very hyperbolic. In particular, we prove the Whitney property (the curvilinear distance to be equivalent to the Euclidean one) of the set D ∩{a₁ = 0, a₂ ≥ −1}.