Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

June Huh (허준이)

Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

2010/7/9 Fri 4PM-5PM

The chromatic polynomial of a graph counts the number of proper colorings of the graph. We give an affirmative answer to the conjecture of Read (1968) and Welsh (1976) 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 show log-concavity of the sequence by answering a question of Trung and Verma on mixed multiplicities of ideals. The conjecture on the chromatic polynomial follows as a special case.

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