Small-span characteristic polynomials of integer symmetric matrices

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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.

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McKee, J. (2010). Small-span characteristic polynomials of integer symmetric matrices. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6197 LNCS, pp. 270–284). https://doi.org/10.1007/978-3-642-14518-6_22

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