Small-span characteristic polynomials of integer symmetric matrices

Abstract

Let f(x)εZ[x] be a totally real polynomial with roots α 1 ≤ α d . The span of f(x) is defined to be α d -α 1. Monic irreducible f(x) of span less than 4 are special. In this paper we give a complete classification of those small-span polynomials which arise as characteristic polynomials of integer symmetric matrices. As one application, we find some low-degree polynomials that do not arise as the minimal polynomial of any integer symmetric matrix: these provide low-degree counterexamples to a conjecture of Estes and Guralnick [6]. © 2010 Springer-Verlag Berlin Heidelberg.