A simplified version of the "Axis of Evil Theorem" for distinct points

TL;DR

This paper simplifies and proves the correctness of an algorithm for computing a linear factorization of the Groebner basis for the ideal of a finite set of distinct points, based on the 'Axis of Evil Theorem.'

Contribution

It provides a finite, correct algorithm for the 'Axis of Evil' factorization, starting from the Groebner escalier, and introduces the 'potential expansion' algorithm for minimal basis computation.

Findings

Algorithm terminates in finite steps

Correctly computes the Groebner basis factorization

Introduces the 'potential expansion' algorithm

Abstract

Given a finite set $\mathbf{X}$ of distinct points, Marinari-Mora's 'Axis of Evil Theorem' states that a combinatorial algorithm and interpolation enable to find a 'linear' factorization for a lexicographical minimal Groebner basis $\mathcal{G}(I(\mathbf{X}))$ of the zerodimensional radical ideal $I(\mathbf{X})$. In this work we provide such algorithm, showing that it ends in a finite number of steps and that it actually provides the correct result. The 'Axis of Evil' algorithm takes as input the monomial basis of the initial ideal $T(I(\mathbf{X}))$ but its starting point is the (finite) Groebner escalier $N$ (obtained via Cerlienco-Mureddu correspondence) so we will also define the `potential expansion' 's algorithm, a combinatorical algorithm which computes the minimal basis from a finite Groebner escalier.