Normaliz: development and long term sustainability

归一化是一个数学研究工具，自1997年以来一直在osnabrck开发。它通过求解方程、不等式和同余的线性丢芬图系统来执行离散凸几何的计算。此外，解可以按度计数，这相当于Hilbert Ehrhart级数的计算。特殊的任务是计算凸多面体的面和点阵点。近年来，正则化的范围通过代数多面体的计算得到了扩展。归一化在数学的许多领域都有应用。它们包括多面体理论、组合代数与几何、代数统计、整数优化、组合拓扑和群论。它也被用于理论物理和数学音乐。normalize是用c++编写的。它适用于Linux、MacOS和MS Windows。对于多项式算法，它使用cocoalition，对于特殊用途，它依赖于其他可选库，如Flint。除了在文件中输入数据之外，它还通过一个非常灵活的模板控制和扩展的c++类库提供访问。normalize算法是为共享内存系统上的并行化而开发的，因此非常强大。normalize的传播得益于它在几个计算机代数系统中的可用性，如CoCoA、GAP、Macaulay2、SageMath和Singular。这个项目的第一个目标是通过进一步的数学上重要和有吸引力的函数来扩展归一化。这包括Gröbner， Markov和Graver基的计算，它们在代数统计中有许多应用。进一步的补充是利用自同构群来计算进一步的不变量，这在某种程度上是一个难题。希尔伯特级数的计算将从单阶级数扩展到多阶级数，特别是在参数问题中有应用。我们还将创建一个面向环代数几何的c++类库，其组合基由有理多面体扇形构成。为了提高计算能力，必须加入异构并行；它还将允许云计算。归一化算法将由线性规划的方法补充，然后也将用于代数多面体。

Normaliz is a tool for mathematical research that has been developed in Osnabrück since 1997. It performs computations in discrete convex geometry by solving linear Diophantine systems of equations, inequalities and congruences. In addition, solutions can be counted by degree, which amounts to the computation of Hilbert Ehrhart series. Special tasks are the computation of facets of convex polytopes and the lattice points. Recently the scope of Normaliz has extended by the computation of algebraic polytopes.Normaliz has found applications in many areas of mathematics. They include polytope theory, combinatorial algebra and geometry, algebraic statistics, integer optimization, combinatorial topology and group theory. It has also been used in theoretical physics and mathematical music.Normaliz is written in C++. It is available for Linux, MacOS and MS Windows. For polynomial arithmetic is uses CoCoALib, and for special purposes it falls back on further optional libraries line Flint. In addition to the input of data in files it offers access via a very flexible template controlled and extensive C++ class library. The algorithms of Normaliz have been developed for parallelization on shared memory systems and are therefore very powerful. The dissemination of Normaliz has profited from its availability in several computer algebra systems like CoCoA, GAP, Macaulay2, SageMath and Singular.The first goal of this project is the extension of Normaliz by further mathematically important and attractive functions. This includes the computation of Gröbner, Markov and Graver bases, which have numerous applications in algebraic statistics. A further addition is the exploitation of automorphism groups for the computation of further invariants, which is to some extent a difficult problem. The computation of Hilbert series will be extended from the singly graded to the multigraded case which in particular has applications in parametric problems. We will also create a C++ class library for toric algebraic geometry whose combinatorial basis is formed by rational polyhedral fans. For the increase of computation power it is necessary to add heterogeneous parallelization; it will also allow cloud computing. The algorithms of Normaliz will be complemented by methods of linear programming, which then will also be available for algebraic polytopes.