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On the discrete Heine-Shephard problem for four lattice polygons

Darren Gerrity, Ivan Soprunov

TL;DR

The paper investigates the discrete Heine–Shephard problem for four planar lattice polytopes by examining the six mixed areas $v_{ij}=\mathop{\mathrm{V}}(P_i,P_j)$ and their place within the discrete Plücker inequalities. It introduces a discrete diagram that records lattice widths and mixed areas, revealing arithmetic constraints absent in the continuous setting. The authors prove that every boundary point of the discrete Plücker set ${\large\text{ Pl}}_{4,2}(\mathbb Z)$ is realizable by four lattice segments, while in the interior they identify both realizable and non-realizable families, including an infinite non-realizable interior family. They also establish a semigroup and saturation structure for possible mixed-area values given fixed widths, and show that lattice arithmetic imposes strict restrictions beyond the Plücker inequalities, highlighting a clear distinction between the continuous and lattice cases with implications for intersection theory and lattice geometry.

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 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.

On the discrete Heine-Shephard problem for four lattice polygons

TL;DR

The paper investigates the discrete Heine–Shephard problem for four planar lattice polytopes by examining the six mixed areas and their place within the discrete Plücker inequalities. It introduces a discrete diagram that records lattice widths and mixed areas, revealing arithmetic constraints absent in the continuous setting. The authors prove that every boundary point of the discrete Plücker set is realizable by four lattice segments, while in the interior they identify both realizable and non-realizable families, including an infinite non-realizable interior family. They also establish a semigroup and saturation structure for possible mixed-area values given fixed widths, and show that lattice arithmetic imposes strict restrictions beyond the Plücker inequalities, highlighting a clear distinction between the continuous and lattice cases with implications for intersection theory and lattice geometry.

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 with prescribed Newton polytopes and generic coefficients. It is known that for arbitrary convex bodies in , 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.
Paper Structure (8 sections, 12 theorems, 55 equations, 9 figures)

This paper contains 8 sections, 12 theorems, 55 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The case of three planar lattice polytopes
  4. The discrete diagram $\left(\mathop{\mathrm{w}}\nolimits_u(P),\mathop{\mathrm{V}}\nolimits(P,Q), \mathop{\mathrm{w}}\nolimits_u(Q)\right)$
  5. The sets $\mathcal{V}^{\,\rm{sf}}_{4,2}(\mathbb Z)$ and ${\large\text{ Pl}}\,_{4,2}(\mathbb Z)$
  6. The boundary of ${\large\text{ Pl}}\,_{4,2}(\mathbb Z)$
  7. Realizable family in the interior of ${\large\text{ Pl}}\,_{4,2}(\mathbb Z)$
  8. Non-realizable family in the interior of ${\large\text{ Pl}}\,_{4,2}(\mathbb Z)$

Key Result

Proposition 2.2

Let $P\in\mathcal{P}(\mathbb Z^2)$. Then

Figures (9)

  • Figure 2.1: The vertices of $P$ and the normalized outer normals of $Q$
  • Figure 4.1: Lattice triangles $P_0$ and $Q_0$
  • Figure 4.2: Lattice triangle $Q$ in Case 1
  • Figure 4.3: Lattice triangle $Q_0$ in Case 2
  • Figure 4.5: Two rational polygons $R_P$ and $R_Q$
  • ...and 4 more figures

Theorems & Definitions (17)

  • Example 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Remark 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Theorem 4.5
  • Remark 4.6
  • ...and 7 more