Full paper in PDF:
$%V. P. Kostov, Root arrangements of hyperbolic polynomial-like
functions,
Rev. Mat. Complut. 19 (2006), no. 1, 197–225.%$
A real polynomial of degree in one real variable is hyperbolic if its roots are all real. A real-valued function is called a hyperbolic polynomial-like function (HPLF) of degree if it has real zeros and vanishes nowhere. Denote by the roots of , , . Then in the absence of any equality of the form
| (1) |
one has
| (2) |
(the Rolle theorem). For (resp. for ) not all arrangements without equalities (1) of real numbers and compatible with (2) are realizable by the roots of hyperbolic polynomials (resp. of HPLFs) of degree and of their derivatives. For and when we show that from the arrangements without equalities (1) and compatible with (2) only are realizable by HPLFs (from which by perturbations of hyperbolic polynomials and none by hyperbolic polynomials).
Key words: hyperbolic polynomial, polynomial-like function, root arrangement, configuration
vector.
2000 Mathematics Subject Classification: 12D10.