The Zero-Free Intervals for Characteristic Polynomials of Matroids

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)P (M, t) > (1 − t) 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) > (t− 1) 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 ].