Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals

TL;DR

This paper introduces a numerical approach to describe positive-dimensional ideals using dual functionals, providing a stable alternative to standard basis algorithms and a stopping criterion for identifying generators.

Contribution

It presents a new stopping criterion based on homogenization for positive-dimensional ideals, improving the computation of dual spaces and Hilbert functions.

Findings

The proposed method guarantees all generators of the initial monomial ideal are found.

It offers a numerically stable alternative to standard basis algorithms.

The approach is applicable to calculating Hilbert functions for positive-dimensional ideals.

Abstract

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.