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Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

Author: June Huh
Journal: J. Amer. Math. Soc. 25 (2012), 907-927
MSC (2010): Primary 14B05, 05B35
Posted: February 8, 2012
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Abstract: The chromatic polynomial $\chi _G(q)$ of a graph counts the number of proper colorings of . We give an affirmative answer to the conjecture of Read and Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. The proof is obtained by identifying $\chi _G(q)$ with a sequence of numerical invariants of a projective hypersurface analogous to the Milnor number of a local analytic hypersurface. As a by-product of our approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor number with the Newton polytope; we also characterize homology classes of $\mathbb{P}^n \times \mathbb{P}^m$ corresponding to subvarieties and answer a question posed by Trung-Verma.

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