On the Annihilator Ideal of an Inverse Form. A Simplification

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.