Computing generating sets of lattice ideals (original) (raw)

In this article, we present a new algorithm for computing generating sets and Gröbner bases of lattice ideals. In contrast to other existing methods, our algorithm starts computing in projected subspaces and then iteratively lifts the results back into higher dimensions, by using a completion procedure, until the original dimension is reached. We give a completely geometric presentation of our Projectand-Lift algorithm and describe also the two other existing main algorithms in this geometric framework. We then give more details on an efficient implementation of this algorithm, in particular on critical-pair criteria specific to lattice ideal computations. Finally, we conclude the paper with a computational comparison of our implementation of the Project-and-Lift algorithm in 4ti2 with algorithms for lattice ideal computations implemented in CoCoA and Singular. Our algorithm outperforms the other algorithms in every single instance we have tried.