PALP: A Package for Analysing Lattice Polytopes with applications to toric geometry

Abstract

We describe our package PALP of C programs for calculations with lattice polytopes and applications to toric geometry, which is freely available on the internet. It contains routines for vertex and facet enumeration, computation of incidences and symmetries, as well as completion of the set of lattice points in the convex hull of a given set of points. In addition, there are procedures specialized to reflexive polytopes such as the enumeration of reflexive subpolytopes, and applications to toric geometry and string theory, like the computation of Hodge data and fibration structures for toric Calabi–Yau varieties. The package is well tested and optimized in speed as it was used for time consuming tasks such as the classification of reflexive polyhedra in 4 dimensions and the creation and manipulation of very large lists of 5-dimensional polyhedra. While originally intended for low-dimensional applications, the algorithms work in any dimension and our key routine for vertex and facet enumeration compares well with existing packages. Program summary Program obtainable form: CPC Program Library, Queen's University of Belfast, N. Ireland Title of program: PALP Catalogue identifier: ADSQ Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSQ Computer for which the program is designed: Any computer featuring C Computers on which it has been tested: PCs, SGI Origin 2000, IBM RS/6000, COMPAQ GS140 Operating systems under which the program has been tested: Linux, IRIX, AIX, OSF1 Programming language used: C Memory required to execute with typical data: Negligible for most applications; highly variable for analysis of large polytopes; no minimum but strong effects on calculation time for some tasks Number of bits in a word: arbitrary Number of processors used: 1 Has the code been vectorised or parallelized?: No Number of bytes in distributed program, including test data, etc.: 138 098 Distribution format: tar gzip file Keywords: Lattice polytopes, facet enumeration, reflexive polytopes, toric geometry, Calabi–Yau manifolds, string theory, conformal field theory Nature of problem: Certain lattice polytopes called reflexive polytopes afford a combinatorial description of a very large class of Calabi–Yau manifolds in terms of toric geometry. These manifolds play an essential role for compactifications of string theory. While originally designed to handle and classify reflexive polytopes, with particular emphasis on problems relevant to string theory applications [M. Kreuzer and H. Skarke, Rev. Math. Phys. 14 (2002) 343], the package also handles standard questions (facet enumeration and similar problems) about arbitrary lattice polytopes very efficiently. Method of solution: Much of the code is straightforward programming, but certain key routines are optimized with respect to calculation time and the handling of large sets of data. A double description method (see, e.g., [D. Avis et al., Comput. Geometry 7 (1997) 265]) is used for the facet enumeration problem, lattice basis reduction for extended gcd and a binary database structure for tasks involving large numbers of polytopes, such as classification problems. Restrictions on the complexity of the program: The only hard limitation comes from the fact that fixed integer arithmetic (32 or 64 bit) is used, allowing for input data (polytope coordinates) of roughly up to 10 9. Other parameters (dimension, numbers of points and vertices, etc.) can be set before compilation. Typical running time: Most tasks (typically: analysis of a four dimensional reflexive polytope) can be perfomed interactively within milliseconds. The classification of all reflexive polytopes in four dimensions takes several processor years. The facet enumeration problem for higher (e.g., 12–20) dimensional polytopes varies strongly with the dimension and structure of the polytope; here PALP's performance is similar to that of existing packages [Avis et al., Comput. Geometry 7 (1997) 265]. Unusual features of the program: None