Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals

Details Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals

2012-01-11

arxiv.org/abs/1201.2242v1

Mathematics

Algebraic Geometry

17 pages, 4 figures

Scientific paper

An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by finding the space of dual functionals that annihilate it, reducing the problem to one of linear algebra. There are several known algorithms for finding the truncated dual up to any specified degree, which is useful for describing zero-dimensional ideals. We present a stopping criterion for positive-dimensional cases based on homogenization that guarantees all generators of the initial monomial ideal are found. This has applications for calculating Hilbert functions.

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