2025/07/04 by Fedor Selyanin, Selyanin, Fedor · 1 citation
Mathematics · #History and Theory of Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2507.03661
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 Arnold's monotonicity problem. The local h^*-polynomial (or ℓ^*-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) ≥ ℓ^*(P;1). The Bk-polytopes are non-trivial examples of thin polytopes. We generalize the Newton number in two independent ways: the ℓ-Newton number and the e-Newton number. The ℓ-Newton number comes from Ehrhart theory, namely, from certain generalizations of Katz-Stapledon decomposition formulas, and its properties are central to our proof that the Bk-polytopes are thin. The e-Newton number is the number of points of zero-dimensional critical complete intersections. Vanishing of the e-Newton number characterizes dual defective sets. Furthermore, the e-Newton number calculates algebraic degrees (such as Maximum Likelihood, Euclidean Distance and Polar degrees). For instance, we show that all known formulas for these algebraic degrees in the Newton non-degenerate case are implied by basic properties of the e-Newton number.