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The Zero-Free Intervals for Characteristic Polynomials of Matroids

Published online by Cambridge University Press:  01 June 1998

HUGH EDWARDS
Affiliation:
Department of Mathematical and Computing Sciences, Goldsmiths College, London SE14 6NW, UK (e-mail: b.jackson@gold.ac.uk)
ROBERT HIERONS
Affiliation:
Department of Mathematical and Computing Sciences, Goldsmiths College, London SE14 6NW, UK (e-mail: b.jackson@gold.ac.uk)
BILL JACKSON
Affiliation:
Department of Mathematical and Computing Sciences, Goldsmiths College, London SE14 6NW, UK (e-mail: b.jackson@gold.ac.uk)

Abstract

Let M be a loopless matroid with rank r and c components. Let P(M, t) be the characteristic polynomial of M. We shall show that (−1)rP(M, t)[ges ](1−t)r for t∈(−∞, 1), that the multiplicity of the zeros of P(M, t) at t=1 is equal to c, and that (−1)r+cP(M, t)[ges ](t−1)r for t∈(1, 32/27]. Using a result of C. Thomassen we deduce that the maximal zero-free intervals for characteristic polynomials of loopless matroids are precisely (−∞, 1) and (1, 32/27].

Type
Research Article
Copyright
1998 Cambridge University Press

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