Searching for Hyperbolic Polynomials with Span Less than 4

Abstract

A monic, irreducible polynomial in one variable having integer coefficients and all real roots deserves particular interest if its roots lie in an interval of length 4 whose end-points are not integers. This follows by some pioneering studies by R. Robinson. Thanks to the crucial support of computers, a number of contributions over the decades settled the existence question for such polynomials up to degree 18. In this article, we find out that almost all of these polynomials can be recovered with algebraic operations from a few polynomials of small degree. Furthermore, a great number of the polynomials discovered by Robinson can be actually obtained as simple linear combinations of Chebyshev polynomials. As a byproduct, we found several families of hyperbolic polynomials related to Salem’s numbers.

Keywords:

• Algebraic integer
• hyperbolic polynomial
• Chebyshev polynomial

MSC (2010):

• 11C08
• 11Y99
• 12D10

Acknowledgements

We are grateful to the anonymous referee for valuable comments and suggestions. This article was prepared with the financial support of Sapienza University of Rome, Progetti di Ateneo.