Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs (Q2892821)

A sequence \(a_0, \dots, a_n\) of real numbers is called log-concave if \(a_{i-1}a_{i+1}\leq a_i^2\) for all \(i, 0<i<n\).NEWLINENEWLINEThe aim of the paper is to answer the following question.NEWLINENEWLINELet \(\chi_G(t)=a_nt^n-a_{n-1}t^{n-1}+\cdots +(-1)^n a_0\) be the chromatic polynomial of a graph \(G\). Is the sequence \(a_0, \dots, a_n\) log-concave? It is proved that for a matroid \(M\) which is representable over a field of characteristic zero the coefficients of the characteristic polynomial of \(M\) form a sign-alternating log-concave sequence of integers with no internal zeros.