Characteristic numbers and chromatic polynomial of a tensor

Abstract

We introduce the characteristic numbers and the chromatic polynomial of a tensor. Our approach generalizes and unifies the chromatic polynomial of a graph and of a matroid, characteristic numbers of quadrics in Schubert calculus, Betti numbers of complements of hyperplane arrangements and Euler characteristic of complements of determinantal hypersurfaces and the maximum likelihood degree for general linear concentration models in algebraic statistics.

Enumerative Geometry Meets Statistics, Combinatorics, and Topology

“…We obtain the Lorentzian polynomial given by deg(∑ 𝑡 𝑖 𝐿 𝑖 ) 𝑎−1 [𝑌 𝐿 ]. In [CM21] the coefficients of this polynomial were introduced and called the characteristic numbers.…”

“…By setting 𝑡 2 = ⋯ = 𝑡 𝑛−1 = 0 we recover the chromatic polynomial of a tensor, defined also in [CM21]. Its coefficients (up to binomial factors) are precisely the multidegrees of the graph of the map inverting matrices 8 from 𝐿.…”