On the Annihilator Ideal of an Inverse Form. A Simplification

TL;DR

This paper simplifies the proof and algorithm for computing the annihilator ideal of inverse forms, removing complex cases and using a more direct approach, resulting in a more efficient and elegant solution.

Contribution

It provides a shorter, more straightforward proof that the intermediate forms form a minimal Groebner basis and introduces a variant of the Berlekamp-Massey algorithm without length change.

Findings

Proved the intermediate forms form a minimal Groebner basis without syzygy polynomials.

Simplified the proof that forms are reduced or reducible, avoiding multiple cases.

Presented a variant of the Berlekamp-Massey algorithm that does not rely on length change.

Abstract

We simplify an earlier paper of the same title by not using syzygy polynomials and by not using a trichotomy of inverse forms. Let $\K$ be a field and $\M=\K[x^{-1},z^{-1}]$ denote Macaulay's $\K[x,z]$ module of inverse polynomials; here $z$ and $z^{-1}$ are homogenising variables. An inverse form $F\in\M$ has a homogeneous annihilator ideal, $\I_F$\,. In an earlier paper we inductively constructed an ordered pair ($f_1$\,,\,$f_2$) of forms in $\K[x,z]$ which generate $\I_F$. We used syzygy polynomials to show that the intermediate forms give a minimal grlex Groebner basis, which can be efficiently reduced. We give a significantly shorter proof that the intermediate forms are a minimal grlex Groebner basis for $\I_F$\,. We also simplify our proof that either $ F$ is already reduced or a monomial of $f_1$ can be reduced by $f_2$\,. The algorithm that computes $f_1\,,f_2$ yields a variant of the Berlekamp-Massey algorithm which does not use the last 'length change' approach of Massey. These new proofs avoid the three separate cases, 'triples' and the technical factorisation of intermediate 'essential' forms. We also show that $f_1,f_2$ is a maximal $\R$ regular sequence for $\I_F$\,, so that $\I_F$ is a complete intersection.