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Papers

Solving Thousand Digit Frobenius Problems Using Grobner Bases
  Journal of Symbolic Computation (January 2008), volume 43, issue 1
  [PDF, PostScript, arXiv, ScienceDirect, BibTeX] download benchmark data
Abstract: A Grobner basis-based algorithm for solving the Frobenius Instance Problem is presented, and this leads to an algorithm for solving the Frobenius Problem that can handle numbers with thousands of digits. Connections to irreducible decompositions and Hilbert functions are also presented.

The Slice Algorithm For Irreducible Decomposition of Monomial Ideals
  To appear in Journal of Symbolic Computation
  [arXiv, BibTeX] download benchmark data
Abstract: Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial ideals. The paper includes experiments showing good performance in practice.

Preprints

The Label Algorithm For Irreducible Decomposition of Monomial Ideals
  [arXiv, BibTeX] download benchmark data
Abstract: Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math, calling for the decomposition of ideals with a large amount of minimal generators. This paper presents an algorithm to compute such decompositions along with benchmarks showing a performance improvement by a factor of up to more than 1000. Performance can be further improved by tailoring the algorithm to specific applications, and some examples of this are presented.

Maximal lattice free bodies, test sets and the Frobenius problem [arXiv, BibTeX]
Joint work with Anders Nedergaard Jensen and Niels Lauritzen.
Abstract: Maximal lattice free bodies are maximal polytopes without interior integral points. Scarf initiated the study of maximal lattice free bodies relative to the facet normals in a fixed matrix. In this paper we give an efficient algorithm for computing the maximal lattice free bodies of an integral matrix A. An important ingredient is a test set for a certain integer program associated with A. This test set may be computed using algebraic methods. As an application we generalize the Scarf-Shallcross algorithm for the three-dimensional Frobenius problem to arbitrary dimension. In this context our method is inspired by the novel algorithm by Einstein, Lichtblau, Strzebonski and Wagon and the Groebner basis approach by Roune.