## Kyoto Journal of Mathematics

### Lattice multipolygons

#### 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 $\mathbb{Z}^{2}$. We first prove a formula on the rotation number of a unimodular sequence in $\mathbb{Z}^{2}$. 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
Revised: 1 July 2016
Accepted: 6 July 2016
First available in Project Euclid: 22 June 2017

https://projecteuclid.org/euclid.kjm/1498096939

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

Mathematical Reviews number (MathSciNet)
MR3725261

Zentralblatt MATH identifier
06825578

#### 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

#### References

• [1] M. Beck and S. Robins,Computing the Continuous Discretely, Undergrad. Texts in Math., Springer, New York, 2007.
• [2] W. Castryck,Moving out the edges of a lattice polygon, Discrete Comput. Geom.47(2012), 496–518.
• [3] R. Diaz and S. Robins,Pick’s formula via the Weierstrass $\wp$-function, Amer. Math. Monthly102(1995), 431–437.
• [4] W. Fulton,An Introduction to Toric Varieties, Ann. of Math. Stud.113, Princeton Univ. Press, Princeton, 1993.
• [5] B. Grünbaum and G. C. Shephard,Pick’s theorem, Amer. Math. Monthly100(1993), 150–161.
• [6] A. Hattori and M. Masuda,Theory of multifans, Osaka J. Math.40(2003), 1–68.
• [7] A. M. Kasprzyk and B. Nill,Reflexive polytopes of higher index and the number $12$, Electron. J. Combin.19(2012), no. 9.
• [8] M. Masuda,Unitary toric manifolds, multi-fans and equivariant index, Tohoku Math. J. (2)51(1999), 237–265.
• [9] T. Oda,Convex Bodies and Algebraic Geometry, Ergeb. Math. Grenzgeb. (3), Springer, Berlin, 1988.
• [10] G. Pick,Geometrisches zur Zahlentheorie, Sitzenber. Lotos (Prague)19(1899), 311–319.
• [11] B. Poonen and F. Rodriguez-Villegas,Lattice polygons and the number $12$, Amer. Math. Monthly107(2000), 238–250.
• [12] D. Repovsh, M. Skopenkov, and M. Tsentsel$^{\prime}$,An elementary proof of the twelve lattice point theorem(Russian), Mat. Zametki77(2005), 117–120; English translation in Math. Notes77(2005), 108–111.
• [13] P. R. Scott,On convex lattice polygons, Bull. Austral. Math. Soc.15(1976), 395–399.
• [14] R. T. Živaljević,Rotation number of a unimodular cycle: An elementary approach, Discrete Math.313(2013), 2253–2261.