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.

摘要

一个具有整数系数和所有实根的变量中的一元不可约多项式如果其根位于长度为 4 且端点不是整数的区间内，则值得特别关注。紧随其后的是 R. Robinson 的一些开创性研究。由于计算机的关键支持，几十年来的许多贡献解决了此类多项式的存在性问题，直到 18 次。在本文中，我们发现几乎所有这些多项式都可以通过几个多项式的代数运算来恢复程度小的。此外，罗宾逊发现的大量多项式实际上可以作为切比雪夫多项式的简单线性组合获得。作为副产品，我们发现了几个与塞勒姆数相关的双曲多项式族。