Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

Details Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

2010-08-27

arxiv.org/abs/1008.4749v3

Mathematics

Algebraic Geometry

Improved readability. Final version, to appear in J. Amer. Math. Soc

Scientific paper

The chromatic polynomial of a graph G counts the number of proper colorings of G. 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. We define a sequence of numerical invariants of projective hypersurfaces analogous to the Milnor number of local analytic hypersurfaces. Then we give a characterization of correspondences between projective spaces up to a positive integer multiple which includes the conjecture on the chromatic polynomial as a special case. As a byproduct of our approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor number with the Newton polytope.

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