Kyoto Journal of Mathematics

Lattice multipolygons

Akihiro Higashitani and Mikiya Masuda

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Abstract

We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice Z2. We first prove a formula on the rotation number of a unimodular sequence in Z2. This formula implies the generalized twelve-point theorem of Poonen and Rodriguez-Villegas. We then introduce the notion of lattice multipolygons, which is a generalization of lattice polygons, state the generalized Pick’s formula, and discuss the classification of Ehrhart polynomials of lattice multipolygons and also of several natural subfamilies of lattice multipolygons.

Article information

Source
Kyoto J. Math., Volume 57, Number 4 (2017), 807-828.

Dates
Received: 28 April 2014
Revised: 1 July 2016
Accepted: 6 July 2016
First available in Project Euclid: 22 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1498096939

Digital Object Identifier
doi:10.1215/21562261-2017-0016

Mathematical Reviews number (MathSciNet)
MR3725261

Zentralblatt MATH identifier
06825578

Subjects
Primary: 05A99: None of the above, but in this section
Secondary: 51E12: Generalized quadrangles, generalized polygons 57R91: Equivariant algebraic topology of manifolds

Keywords
lattice polygon twelve-point theorem Pick’s formula Ehrhart polynomial toric topology

Citation

Higashitani, Akihiro; Masuda, Mikiya. Lattice multipolygons. Kyoto J. Math. 57 (2017), no. 4, 807--828. doi:10.1215/21562261-2017-0016. https://projecteuclid.org/euclid.kjm/1498096939


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