Consider a polynomial f with a convenient Newton polytope P and generic complex coefficients. By the global version of the Kouchnirenko formula, the hypersurface \f = 0\ Cⁿ has the homotopy type of a bouquet of (n-1) -spheres, and the number of spheres is given by a certain alternating sum of volumes, called the Newton number ν (P). Using the Furukawa-Ito classification of dual defective sets, we classify convenient Newton polytopes with vanishing Newton numbers as certain Cayley sums called Bₖ-polytopes. These Bₖ-polytopes generalize the B₁- and B₂-facets appearing in the local monodromy conjecture in the Newton non-degenerate case. Our classification provides a partial solution to the Arnold's monotonicity problem. The local h^*-polynomial (or ^*-polynomial) is a natural invariant of lattice polytopes that refines the h^*-polynomial coming from Ehrhart theory. We obtain decomposition formulas for the Newton number, for instance, prove the inequality ν (P) ^* (P;1). The Bₖ-polytopes are non-trivial examples of thin polytopes. We generalize the Newton number in two independent ways: the -Newton number and the e-Newton number. The -Newton number comes from Ehrhart theory, namely, from certain generalizations of Katz-Stapledon decomposition formulas. It is the main ingredient in the proof of the thinness of the Bₖ-polytopes. The e-Newton number is the number of points of zero-dimensional critical complete intersections. Vanishing of the e-Newton number characterizes the dual defective sets. The e-Newton number calculates the algebraic degrees (Maximum Likelihood, Euclidean Distance, and Polar degrees). For instance, we show that all the known formulas for the algebraic degrees in the Newton non-degenerate case are implied by basic properties of the e-Newton number.
Fedor Selyanin (Fri,) studied this question.