Journal of Symbolic Computation - Special issue on applications of the Gröbner basis method
The Construction of Multivariate Polynomials with Preassigned Zeros
EUROCAM '82 Proceedings of the European Computer Algebra Conference on Computer Algebra
A note on the complexity of constructing Gröbner-Bases
EUROCAL '83 Proceedings of the European Computer Algebra Conference on Computer Algebra
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Modular algorithms for computing Gröbner bases
Journal of Symbolic Computation
Solving the Likelihood Equations
Foundations of Computational Mathematics
Computing gröbner bases for vanishing ideals of finite sets of points
AAECC'06 Proceedings of the 16th international conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Complexity of Comparing Monomials and Two Improvements of the Buchberger-Möller Algorithm
Mathematical Methods in Computer Science
A bivariate preprocessing paradigm for the Buchberger-Möller algorithm
Journal of Computational and Applied Mathematics
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A contemporary and exciting application of Gröbner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to tens of points, while the number of genes or variables is potentially in the thousands. As such data sets vastly underdetermine the biological network, many models may fit the same data and reverse engineering programs often require the use of methods for choosing parsimonious models. Gröbner bases have recently been employed as a selection tool for polynomial dynamical systems that are characterized by maps in a vector space over a finite field. While there are numerous existing algorithms to compute Gröbner bases, to date none has been specifically designed to cope with large numbers of variables and few distinct data points. In this paper, we present an algorithm for computing Gröbner bases of zero-dimensional ideals that is optimized for the case when the number m of points is much smaller than the number n of indeterminates. The algorithm identifies those variables that are essential, that is, in the support of the standard monomials associated to a polynomial ideal, and computes the relations in the Gröbner basis in terms of these variables. When n is much larger than m, the complexity is dominated by nm3. The algorithm has been implemented and tested in the computer algebra system Macaulay 2. We provide a comparison of its performance to the Buchberger-Möller algorithm, as built into the system.