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Classifying rational polygons with small denominator and few interior lattice points

Martin Bohnert, Justus Springer

Abstract

We present algorithms for classifying rational polygons with fixed denominator and number of interior lattice points. Our approach is to first describe maximal polygons and then compute all subpolygons, where we eliminate redundancy by a suitable normal form. Executing our classification, we obtained a dataset of several billions of polygons covering a wide variety of cases.

Classifying rational polygons with small denominator and few interior lattice points

Abstract

We present algorithms for classifying rational polygons with fixed denominator and number of interior lattice points. Our approach is to first describe maximal polygons and then compute all subpolygons, where we eliminate redundancy by a suitable normal form. Executing our classification, we obtained a dataset of several billions of polygons covering a wide variety of cases.

Paper Structure

This paper contains 10 sections, 19 theorems, 39 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Prerequesites for classifying rational polygons
  3. Normal forms
  4. Subpolygons
  5. Maximal polygons
  6. Rational polygons in R x [-1,1] or with collinear interior points
  7. Rational polygons without interior lattice points
  8. Rational polygons with exactly one interior lattice point and LDP polygons
  9. Half-integral polygons
  10. The general case

Key Result

Theorem 1.1

The numbers of $k$-rational polygons with $i$ interior lattice points is given by the following table. Each cell contains the number of maximal polygons on top, the number of distinct Ehrhart quasipolynomials in the middle and the total number of polygons below. For $i = 0$, only the polygons which

Figures (10)

  • Figure 1: All rational polygons in $\mathop{\mathrm{\mathds{R}}}\nolimits \times [-1,1]$ with $i$ interior lattice points can be realized in $\mathop{\mathrm{\mathds{R}}}\nolimits \times [-1,0] \cup T$
  • Figure 2: All rational polygons without interior lattice points can be realized in $A \cup \mathop{\mathrm{\mathds{R}}}\nolimits\times [0,1]\cup B$.
  • Figure 3: All four $2$-maximal rational polygons without interior integral points, see also AKW17. All of them can be realized in $\mathop{\mathrm{\mathds{R}}}\nolimits \times [-1,1]$.
  • Figure 4: All fourteen $3$-maximal rational polygons without interior integral points. The two rightmost polygons in the lower row are the only ones that cannot be realized in $\mathop{\mathrm{\mathds{R}}}\nolimits \times [-1,1]$.
  • Figure 5: All thirtynine $4$-maximal rational polygons without interior lattice points. The 24 polygons in the upper three rows can be realized in $\mathop{\mathrm{\mathds{R}}}\nolimits \times [-1,1]$. The remaining 15 polygons cannot.
  • ...and 5 more figures

Theorems & Definitions (49)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Remark 2.8
  • ...and 39 more