Vector space bases associated to vanishing ideals of points

TL;DR

This paper explores four methods for constructing vector space bases related to vanishing ideals of points, providing algorithms for normal forms, analyzing complexity, and applying these to gene network reverse engineering.

Contribution

It introduces four novel constructions of bases for vanishing ideals, along with efficient computation methods and complexity bounds, enhancing algebraic approaches to biological network inference.

Findings

Developed four new basis constructions for vanishing ideals

Provided algorithms for computing normal forms efficiently

Applied methods to improve gene regulatory network reverse engineering

Abstract

We discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give new complexity bounds. As an application, we improve the computational algebra approach to the reverse engineering of gene regulatory networks.