Normaliz is a program for the computation of Hilbert bases of rational cones and the normalizations of affine monoids. It may also be used for solving diophantine linear systems. In this paper we present the algorithms implemented in the program.

We describe the use of pyramid decomposition in Normaliz, a software tool for the computation of Hilbert bases and enumerative data of rational cones and affine monoids. Pyramid decomposition in connection with efficient parallelization and streamlined evaluation of simplicial cones has enabled Normaliz to process triangulations of size ≈ 5 · 10 11 that arise in the computation of Ehrhart series related to the theory of social choice.

Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley-Reisner and affine monoid algebras. The main goal of this article is to unify parts of the theories of Stanley-Reisner and affine monoid algebras. We consider (non-pure) shellable fan's and the Cohen-Macaulay property. Moreover, we study the local cohomology, the canonical module and the Gorenstein property of a toric face ring.

We describe several experimental results obtained in four candidates social choice elections. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality voting with runoff. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes and counting functions encoded as Ehrhart series of polytopes.2010 Mathematics Subject Classification. 52B20, 91B12.

In the first part of this paper we introduce a method for computing Hilbert decompositions (and consequently the Hilbert depth) of a finitely generated multigraded module M over the polynomial ring K [X 1 , . . . , X n ] by reducing the problem to the computation of the finite set of the new defined Hilbert partitions. In the second part we show how Hilbert partitions may be used for computing the Stanley depth of the module M. In particular, we answer two open questions posed by Herzog in [8].