Newton numbers, vanishing polytopes and algebraic degrees

TL;DR

This paper classifies Newton polytopes with vanishing Newton numbers, introduces generalized Newton numbers, and links these concepts to algebraic degrees and the topology of hypersurfaces, advancing understanding in algebraic geometry and polytope theory.

Contribution

It provides a classification of polytopes with zero Newton number, introduces new variants of Newton numbers, and connects these to algebraic degrees and topological invariants.

Findings

Classified vanishing Newton number polytopes as $B_k$-polytopes.

Established inequalities relating Newton number and $h^*$-polynomial.

Connected $e$-Newton number to algebraic degrees like ML and Euclidean Distance.

Abstract

Consider a polynomial $f$ with a convenient Newton polytope $P$ and generic complex coefficients. By the global version of the Kouchnirenko formula, the hypersurface $\{f = 0\} \subset \mathbb{C}^n$ 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 $\nu(P)$. Using the Furukawa-Ito classification of dual defective sets, we classify convenient Newton polytopes with vanishing Newton numbers as certain Cayley sums called $B_k$-polytopes. These $B_k$-polytopes generalize the $B_1$- and $B_2$-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 $\ell^*$-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 $\nu(P) \ge \ell^*(P;1)$. The $B_k$-polytopes are non-trivial examples of thin polytopes. We generalize the Newton number in two independent ways: the $\ell$-Newton number and the $e$-Newton number. The $\ell$-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 $B_k$-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.