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

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.