Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

By June Huh. 2010

237 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/67ea06d189a6a03df80951e0de2241e16be87994

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Citations in the corpus

Algebraic and Geometric Methods in Enumerative Combinatorics (2014, methodology): “It was conjectured by Rota in 1970 [Rot71], and recently proved by Huh [Huh12] for fields of characteristic 0 and by Huh and Katz [HK12] for arbitrary fields, drawing from toric geometry, tropical geometry, and matroid theory.”
Lagrangian geometry of matroids (2020, background): “The log-concavity of the f -vector of the broken circuit complex was proved in [Huh12] for matroids representable over a field of characteristic 0.”
h-Vectors of matroids and logarithmic concavity (2012, methodology): “Other illustrations can be found in [Huh12a, HK12, Len12].”
The maximum likelihood degree of a very affine variety (2012, methodology): “It can be shown that some multiple of ξ is the fundamental class of an irreducible subvariety if and only if the ei form a log-concave sequence of nonnegative integers with no internal zeros [Huh12a, Theorem 21].”
COMBINATORIAL APPLICATIONS OF THE HODGE–RIEMANN RELATIONS (2017, methodology): “References Karim Adiprasito, June Huh, and Eric Katz (Nov. 2015).”
Grothendieck Classes and Chern Classes of Hyperplane Arrangements (2011, methodology): “ S: Xn i=0 ˙ i \[Pn i] = [P n] s(S;P ) : This equality should be compared with Huh’s formula expressing the Betti numbers of the complement as mixed multiplicities; see the proof of Corollary 16 in [Huh]. It is natural to ask whether any such relation may hold for more general hypersurfaces. Finall”
Matroid Theory for Algebraic Geometers (2014, background): “This conjecture was proved by Huh in 2010 [44].”
Unimodality, log-concavity, real-rootedness and beyond (2015, background): “e matroids. Conjecture 4.4 (Mason). Let M be a matroid and n= f1(M). The following sequences are log{concave: ff k(M)g r(M) k=0 ; fk!f k(M)g r(M) k=0 ; and ˆ f k(M)= n k ˙ r(M) k=0 : The proofs in [61,62] use involved algebraic machinery which falls beyond the scope of this survey. It is unclear if”
Hodge theory for combinatorial geometries (2015, methodology): “…by the linear forms ∑ i1∈F xF − ∑ i2∈F xF , i1 and i2 are distinct elements of the ground set E. Conjecture 1.1 was proved for matroids realizable over C in [Huh12] by relating wk(M) to the Milnor numbers of a hyperplane arrangement realizing M over C. Subsequently in [HK12], using the…”