Small-Span Characteristic Polynomials of Integer Symmetric Matrices
نویسنده
چکیده
Let f(x) ∈ Z[x] be a totally real polynomial with roots α1 ≤ . . . ≤ αd. The span of f(x) is defined to be αd − α1. 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 lowdegree counterexamples to a conjecture of Estes and Guralnick [6].
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