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Linear bounds on characteristic polynomials of matroids



Let χ(t) = a0tna1tn−1 + ⋯ + (−1)rartnr be the chromatic polynomial of a graph, the characteristic polynomial of a matroid, or the characteristic polynomial of an arrangement of hyperplanes. For any integer k = 0, 1, …, r and real number xkr − 1, we obtain a linear bound of the coefficient sequence, that is

\begin{align*} {r+x\choose k}\leqslant \sum_{i=0}^{k}a_{i}{x\choose k-i}\leqslant {m+x\choose k}, \end{align*}
where m is the size of the graph, matroid, or hyperplane arrangement. It extends Whitney’s sign-alternating theorem, Meredith’s upper bound theorem, and Dowling and Wilson’s lower bound theorem on the coefficient sequence. In the end, we also propose a problem on the combinatorial interpretation of the above inequality.



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This work was supported by the National Natural Science Foundation of China (11871204).



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Linear bounds on characteristic polynomials of matroids



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