Small-Span Characteristic Polynomials of Integer Symmetric Matrices

  • James McKee
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)


Let f(x) ∈ Z[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].


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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • James McKee
    • 1
  1. 1.Department of MathematicsRoyal Holloway, University of LondonEgham Hill, Egham, SurreyEngland, UK

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