On the discrete Heine-Shephard problem for four lattice polygons

TL;DR

This paper investigates the limitations of Plücker-type inequalities in characterizing lattice polytopes' intersection properties, revealing additional arithmetic constraints in the discrete setting.

Contribution

It demonstrates that classical inequalities do not fully describe lattice polytopes' intersection behavior, highlighting new arithmetic constraints in the discrete case.

Findings

Plücker-type inequalities are insufficient for lattice polytopes.

Additional arithmetic constraints affect mixed areas of lattice polytopes.

The discrete diagram reveals these constraints through lattice widths and mixed areas.

Abstract

We study the set of square-free parts of volume polynomials associated with four planar lattice polytopes. This is motivated by the problem of describing possible pairwise intersection numbers of four curves in $(\mathbb{C}^*)^2$ with prescribed Newton polytopes and generic coefficients. It is known that for arbitrary convex bodies in $\mathbb{R}^2$, the corresponding square-free polynomials are characterized by the Pl\"ucker-type inequalities. We show that this characterization fails in the lattice setting: the interior of the space defined by the Pl\"ucker-type inequalities contains integer polynomials that are and are not realizable by lattice polytopes. This phenomenon arises from additional arithmetic constraints on the mixed areas of lattice polytopes. These constraints become apparent when we study a "discrete diagram", which maps a pair of planar lattice polytopes to their mixed area together with their lattice widths in a given direction.