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Is there a smooth lattice polytope which does not have the integer decomposition property?

Johannes Hofscheier, Alexander Kasprzyk

TL;DR

The paper examines Tadao Oda's Oberwolfach question on whether smooth lattice polytopes necessarily have the integer decomposition property (IDP). It connects lattice-point enumeration to Ehrhart theory, using Pick's Theorem in 2D as a motivating example and introducing the cone construction $C_P$ to formalize IDP. It establishes that unimodular simplices and all 2D lattice polygons have IDP via unimodular coverings, while higher dimensions admit counterexamples to naive coverings, exemplified by Reeve tetrahedra. The discussion then centers on the smooth case, where a positive answer is conjectured but remains open, highlighting deep links between combinatorial geometry and lattice point decompositions with implications for discrete and algebraic geometry.

Abstract

We introduce Tadao Oda's famous question on lattice polytopes which was originally posed at Oberwolfach in 1997 and, although simple to state, has remained unanswered. The question is motivated by a discussion of the two-dimensional case - including a proof of Pick's Theorem, which elegantly relates the area of a lattice polygon to the number of lattice points it contains in its interior and on its boundary.

Is there a smooth lattice polytope which does not have the integer decomposition property?

TL;DR

The paper examines Tadao Oda's Oberwolfach question on whether smooth lattice polytopes necessarily have the integer decomposition property (IDP). It connects lattice-point enumeration to Ehrhart theory, using Pick's Theorem in 2D as a motivating example and introducing the cone construction to formalize IDP. It establishes that unimodular simplices and all 2D lattice polygons have IDP via unimodular coverings, while higher dimensions admit counterexamples to naive coverings, exemplified by Reeve tetrahedra. The discussion then centers on the smooth case, where a positive answer is conjectured but remains open, highlighting deep links between combinatorial geometry and lattice point decompositions with implications for discrete and algebraic geometry.

Abstract

We introduce Tadao Oda's famous question on lattice polytopes which was originally posed at Oberwolfach in 1997 and, although simple to state, has remained unanswered. The question is motivated by a discussion of the two-dimensional case - including a proof of Pick's Theorem, which elegantly relates the area of a lattice polygon to the number of lattice points it contains in its interior and on its boundary.
Paper Structure (7 sections, 2 theorems, 16 equations, 9 figures)

This paper contains 7 sections, 2 theorems, 16 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Proof of Pick's Theorem
  3. The base case
  4. The inductive step
  5. Oda's Oberwolfach question
  6. The integer decomposition property
  7. The smooth case

Key Result

Theorem 1

There exists a polynomial $f$ of degree $\dim(P)$ with rational coefficients such that $L_P(m) = f(m)$ for all $m \in \mathbb{Z}_{\ge0}$. Furthermore, the leading coefficient of $f$ coincides with the Euclidean volume $\mathop{\mathrm{vol}}\nolimits(P)$ of $P$.

Figures (9)

  • Figure 1: Interpreting monomials in three variables $x,y$, and $z$ as lattice points in a triangle, for (a) degree one, (b) degree two, and (c) degree three.
  • Figure 2: Computing the area of a lattice polygon by Pick's Theorem. There are 23 interior lattice points and 16 boundary lattice points.
  • Figure 3: Tiling the plane with translations of a parallelogram which comes from an empty triangle with vertices $\bm{v}_1,\bm{v}_2,\bm{v}_3$.
  • Figure 4: A lattice polygon \ref{['subfig:split-in-three-left']} with exactly three boundary lattice points is cut \ref{['subfig:split-in-three-right']} into three subpolygons $P_1,P_2$, and $P_3$. Here, $b_1=4$, $b_2=3$, $b_3=2$ and $i_1=1$, $i_2=i_3=0$.
  • Figure 5: A lattice polygon \ref{['subfig:split-in-two-left']} with four or more lattice points on its boundary can be split \ref{['subfig:split-in-two-right']} into two subpolygons $P_1$ and $P_2$. Here, $i_1=4$, $i_2=14$, $b_1=7$, $b_2=11$, and $i=5$.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Theorem 1: see Ehrhart62Ehrhart67
  • Theorem 2: Pick's Theorem
  • Example 1
  • Example 2
  • Example 3
  • Example 4