Computing gröbner bases for vanishing ideals of finite sets of points

Authors:
Jeffrey B. Farr;Shuhong Gao
Affiliations:
Simon Fraser University, Burnaby, B.C., Canada;Clemson University, Clemson, SC
Venue:
AAECC'06 Proceedings of the 16th international conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Year:
2006

Citing 7
Cited 2

Efficient computation of zero-dimensional Gro¨bner bases by change of ordering

Journal of Symbolic Computation
From algebraic sets to monomial linear bases by means of combinatorial algorithms

Proceedings of the 4th conference on Formal power series and algebraic combinatorics
Decoding of Reed Solomon codes beyond the error-correction bound

Journal of Complexity
Computing ideals of points

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
Improved decoding of Reed-Solomon and algebraic-geometry codes

IEEE Transactions on Information Theory
Algebraic soft-decision decoding of Reed-Solomon codes

IEEE Transactions on Information Theory

Computing Gröbner bases of ideals of few points in high dimensions

ACM Communications in Computer Algebra
Some combinatorial applications of Gröbner bases

CAI'11 Proceedings of the 4th international conference on Algebraic informatics

Quantified Score

Hi-index	0.00

Visualization

Abstract

We present an algorithm to compute a Gröbner basis for the vanishing ideal of a finite set of points in an affine space. For distinct points the algorithm is a generalization of univariate Newton interpolation. Computational evidence suggests that our method compares favorably with previous algorithms when the number of variables is small relative to the number of points. We also present a preprocessing technique that significantly enhances the performance of all the algorithms considered. For points with multiplicities, we adapt our algorithm to compute the vanishing ideal via Taylor expansions.