Normaliz: development and long term sustainability

Normaliz是自1997年以来在奥斯纳布吕克开发的一种数学研究工具。它通过求解线性丢番图方程、不等式和同余系统来执行离散凸几何中的计算。此外，解可以按次数计数，这相当于希尔伯特·埃哈特级数的计算。特殊任务是计算凸多面体的面片和格点。最近，通过计算代数多面体，Normaliz的范围得到了扩展。Normaliz在数学的许多领域都有应用。它们包括多面体理论、组合代数和几何、代数统计、整数优化、组合拓扑学和群论。它还被用于理论物理和数学音乐。Normaliz是用C++编写的。它可用于Linux、MacOS和MS Windows。对于多项式算术，它使用CoCoALib，而对于特殊用途，它依赖于更多可选的库行Flint。除了在文件中输入数据之外，它还提供了通过非常灵活的模板控制和广泛的C++类库进行访问的功能。Normaliz的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。Normaliz的推广得益于它在几个计算机代数系统中的应用，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重要和吸引人的函数来扩展Normaliz。这包括Gröbner基、马尔可夫基和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困难的问题。希尔伯特级数的计算将从单分次扩展到多分次情况，特别是在参数问题中的应用。我们还将创建一个C++类库，用于环形代数几何，其组合基由有理多面体扇形形成。为了增加计算能力，有必要增加异构性并行；它还将允许云计算。Normaliz的算法将得到线性规划方法的补充，然后线性规划方法也将适用于代数多面体。

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.