RationalPolygons.jl

RationalPolygons.jl is a pure Julia package for computations with rational convex polygons. It implements

counting lattice points, Ehrhart Theory, normal forms, automorphism groups, computation of subpolygons,
various classification algorithms for integral and rational polygons.

I have written RationalPolygons.jl in the span of about six months while working on a joint project with Martin Bohnert [1]. Its main purpose is to provide reference implementations of the classification algorithms developed in our paper. However, it also implements many more basic algorithms for computations with rational polygons that I believe might be useful in other projects, hence I have documented them here. RationalPolygons.jl does not make use of any external computer algebra system but implements all necessary algorithms, including two-dimensional euclidian geometry, from scratch in pure Julia. This allows for quite good performance, with computations involving billions of polygons being feasable on a personal computer.

Quick start

julia> using RationalPolygons
julia> P = convex_hull(LatticePoint{Int}[(1,0),(0,1),(-1,1),(-1,0),(0,-1),(1,-1)])Rational polygon of rationality 1 with 6 vertices.
julia> plot_polygon(P)ERROR: UndefVarError: plot_polygon not defined

julia> number_of_interior_lattice_points(P)1
julia> number_of_boundary_lattice_points(P)6
julia> euclidian_area(P)3//1
julia> ehrhart_quasipolynomial(P)1×3 Matrix{Int64}:
 6  6  2
julia> affine_automorphism_group(P)D6
julia> is_ldp(P)true
julia> dual(P)Rational polygon of rationality 1 with 6 vertices.
julia> are_affine_equivalent(P, dual(P))true
julia> gorenstein_index(P)1

[1]: M. Bohnert and J. Springer. Classifying rational polygons with small denominator and few interior lattice points (2024), arXiv:2410.17244 [math.CO].
[2]: J. Springer. The Picard index of a surface with torus action. Collectanea Mathematica (2024).
[3]: D. Haettig, B. Hafner, J. Hausen and J. Springer. Del Pezzo surfaces of Picard number one admitting a torus action (2022), arXiv:2207.14790 [math.AG].
[4]: D. Haettig, J. Hausen and J. Springer. Classifying log del Pezzo surfaces with torus action (2023), arXiv:2302.03095 [math.AG].
[5]: M. Bohnert. Area bounds for planar convex bodies containing a fixed number of interior integral points (2023), arXiv:2305.11485 [math.MG].
[6]: D. A. Cox, J. B. Little and H. K. Schenck. Toric varieties. Vol. 124 of Graduate Studies in Mathematics (American Mathematical Society, Providence, RI, 2011); p. xxiv+841.
[7]: R. J. Koelman. The number of moduli of families of curves on toric surfaces. Ph.D. Thesis, University of Nijmegen (1991).
[8]: W. Castryck. Moving out the edges of a lattice polygon. Discrete Comput. Geom. 47, 496–518 (2012).
[9]: G. Brown and A. M. Kasprzyk. Small polygons and toric codes. J. Symbolic Comput. 51, 55–62 (2013).
[10]: A. M. Kasprzyk, M. Kreuzer and B. Nill. On the combinatorial classification of toric log del Pezzo surfaces. LMS J. Comput. Math. 13, 33–46 (2010).
[11]: A. Bäuerle. Sharp volume and multiplicity bounds for Fano simplices (2023), arXiv:2308.12719 [math.CO].