Journal of Symbolic Computation (January 2008), volume 43, issue 1
arXiv BibTeX PDF ScienceDirect Benchmark data
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.

Journal of Symbolic Computation (April 2009), volume 44, issue 4
arXiv BibTeX PDF ScienceDirect Benchmark data
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.

Proceedings of the 2010 international symposium on Symbolic and algebraic computation
BibTeX PDF ACM Digital Library
Abstract: We present an algorithm for computing the corners of a monomial ideal. The corners are a set of multidegrees that support the numerical information of a monomial ideal such as Betti numbers and Hilbert-Poincaré series. We show an experiment using corners to compute Hilbert-Poincaré series of monomial ideals with favorable results.

arXiv PDF ACM Digital Library Source code
Joint work with Michael Stillman.
We report on our experiences exploring state of the art Groebner basis computation. We investigate signature based algorithms in detail. We also introduce new practical data structures and computational techniques for use in both signature based Groebner basis algorithms and more traditional variations of the classic Buchberger algorithm. Our conclusions are based on experiments using our new freely available open source standalone C++ library.

PDF
An accessible three-page account of Faugere's F4 algorithm, which speeds up Groebner basis computation using linear algebra.

Postscript
An introduction to Matroid theory in Danish.