Normaliz: development and long term sustainability

• Applicant: Professor Dr. Winfried Bruns
• Subject Area: Mathematics
• Term: from 2020 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 440345850

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.

• DFG Programme: Research Grants