Normaliz: Algorithms for Affine Monoids and Rational Cones

Normaliz is a program for solving linear systems of inequalities. In this paper we present the algorithms implemented in the program, starting with version 2.0

The subdivision of large simplicial cones in Normaliz

Normaliz is an open-source software for the computation of lattice points in rational polyhedra, or, in a different language, the solutions of linear diophantine systems. The two main computational goals are (i) finding a system of generators of the set of lattice points and (ii) counting elements degree-wise in a generating function, the Hilbert Series. In the homogeneous case, in which the polyhedron is a cone, the set of generators is the Hilbert basis of the intersection of the cone and the lattice, an affine monoid. We will present some improvements to the Normaliz algorithm by subdividing simplicial cones with huge volumes. In the first approach the subdivision points are found by integer programming techniques. For this purpose we interface to the integer programming solver SCIP to our software. In the second approach we try to find good subdivision points in an approximating overcone that is faster to compute.Comment: To appear in the proceedings of the ICMS 2016, published by Springer as Volume 9725 of Lecture Notes in Computer Science (LNCS

The Steinberg group of a monoid ring, nilpotence, and algorithms

For a regular ring R and an affine monoid M the homotheties of M act nilpotently on the Milnor unstable groups of R[M]. This strengthens the K_2 part of the main result of [G5] in two ways: the coefficient field of characteristic 0 is extended to any regular ring and the stable K_2-group is substituted by the unstable ones. The proof is based on a polyhedral/combinatorial techniques, computations in Steinberg groups, and a substantially corrected version of an old result on elementary matrices by Mushkudiani [Mu]. A similar stronger nilpotence result for K_1 and algorithmic consequences for factorization of high Frobenius powers of invertible matrices are also derived.Comment: final version, to appear in Journal of Algebr