The maximum likelihood degree of a very affine variety (Q2852239)

Let \(U\) be the complement of \(n\) hyperplanes in \(\mathbb C^r\) defined by linear polynomials \(f_1,\dots,f_n\). The function \(\phi_U=\prod_{i=1}^n f_i^{u_i}\) with \(u_i\in \mathbb Z\) is a holomorphic function on \(U\). A hyperplane arrangement is said to be essential if the lowest-dimensional intersections of the hyperplanes are points. Varchenko conjectured that if the hyperplane arrangement is essential and the exponents \(u_i\) are sufficiently general, thenNEWLINENEWLINE(i) \(\phi_U\) has only finitely many critical points in \(U\);NEWLINENEWLINE(ii) All the critical points of \(\phi_U\) are nondegenerate;NEWLINENEWLINE(iii) The number of critical points is equal to the signed Euler characteristic \((-1)^r\chi(U)\).NEWLINENEWLINEThe conjecture was proved by Varchenko for real arrangements [\textit{A. Varchenko}, Compos. Math. 97, No. 3, 385--401 (1995; Zbl 0842.17044)], and by Orlik and Terao in general [\textit{P. Orlik} and \textit{H. Terao}, Invent. Math. 120, No. 1, 1--14 (1995; Zbl 0934.32020)]. The aim of the paper under review is to extend the theorem of Orlik and Terao. The extension is motivated by the problem of maximum likelihood estimation in algebraic statistics. In this context the maximum likelihood degree of \(U\) is defined to be precisely the number of critical points of \(\phi_U\) with sufficiently general exponents \(u_i\). An irreducible algebraic variety is very affine if it is isomorphic to a closed subvariety of an algebraic torus. For example a complement of a hyperplane arrangement is very affine if and only if the hyperplane arrangement is essential.NEWLINENEWLINEIn the paper under review Varchenko's conjecture is proved for very affine varieties; see Theorem 1. Another generalization of the theorem of Orlik and Terao is given in terms of the variety of critical points. Specifically, Theorem 3.8 relates the variety of critical points in \(U\) to the Chern-Schwartz-MacPherson class of [\textit{R. D. MacPherson}, Ann. Math. (2) 100, 423--432 (1974; Zbl 0311.14001)].NEWLINENEWLINELet \(g\) be a nonzero Laurent polynomial with Newton polytope \(\Delta_g\). Set \(U=\{ g=0 \}\subseteq (\mathbb C^{*})^n\). It is shown that the variety of critical points in \(U\) is determined by the Newton polytope \(\Delta_g\); see Theorem 4.1. This gives a formula for the Chern-Schwartz-MacPherson class in terms of the Newton polytope. Some applications to classical projective geometry of homaloidal polynomials are given.