On the discrete Heine-Shephard problem for four lattice polygons

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 Plücker-type inequalities. We show that this characterization fails in the lattice setting: the interior of the space defined by the Plücker-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.

💡 Research Summary

The paper investigates the set of square‑free parts of volume polynomials that arise from four planar lattice polygons. This problem originates from algebraic geometry: given four Laurent polynomials in two variables with prescribed Newton polygons, the pairwise intersection numbers of the corresponding curves in ((\mathbb C^*)^2) equal twice the mixed area of the two Newton polygons. For arbitrary convex bodies in (\mathbb R^2) the mixed areas satisfy three Plücker‑type quadratic inequalities, and these inequalities together with non‑negativity completely describe the possible 6‑tuples ((v_{12},\dots ,v_{34})) of mixed areas (Heine–Shephard theory).

When the bodies are restricted to lattice polygons, the situation changes. The mixed area of two lattice polygons is always a half‑integer, so the obvious integrality condition (2v_{ij}\in\mathbb Z_{\ge0}) must hold. The authors show that this is not sufficient: inside the polyhedral cone defined by the Plücker inequalities there exist integer points that are realizable by lattice polygons and integer points that are not. The obstruction is arithmetic rather than algebraic and is revealed by studying a “discrete diagram”.

For a fixed primitive direction (u\in\mathbb Z^2) the map
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