離散幾何構造セミナー：Log-concavity of chromatic and characteristic polynomials (Survey)

開催日時

2018年3月13日 10時30分 ～ 2018年3月13日 15時00分

場所

3-413

講演者

吉永 正彦（北海道大学）

DGS Seminar 014

The first talk begins with recalling the notions of the chromatic polynomial of a graph and the characteristic polynomial of a hyperplane arrangement. Then we state the log-concavity conjecture for the coefficients of these polynomials which were recently proved by June Huh. In the second part, we recall De Concini-Procesi’s Wonderful compactification and its cohomology (Chow) ring. Then we give a geometric proof for a formula that expresses coefficients of the characteristic polynomial as intersection numbers of certain cycles (Prop 9.5 in [2]). Applying (classical) Hodge-Riemann relation, we obtain log-concavity. (The basic idea of this talks is to prove main results of [1] using the strategy of [2]. We will not discuss main part of [2]. From matroidal point of view, this talk focuses only on matroids realizable over C.)

References:
[1] June Huh, Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, Journal of the American Mathematical Society 25 (2012), 907-927. arXiv:1008.4749
[2] Karim Adiprasito, June Huh, Eric Katz, Hodge Theory for Combinatorial Geometries. arXiv:1511.02888