On the annihilator ideal of an inverse form: addendum

Abstract

We improve results and proofs of our earlier paper and show that the ideal in question is a complete intersection. Let ${\mathbb {K}}$ be a field and $\mathrm {M}={\mathbb {K}}[x^{-1},z^{-1}]$ denote the ${\mathbb {K}}[x,z]$ submodule of Macaulay’s inverse system ${\mathbb {K}}[[x^{-1},z^{-1}]]$. We regard $z\in {\mathbb {K}}[x,z]$ and $z^{-1}\in \mathrm {M}$ as homogenising variables. An inverse form $F\in \mathrm {M}$ has a homogeneous annihilator ideal ${\mathcal {I}}_F$ . In our earlier paper we inductively constructed an ordered pair ($f_1$ , $f_2$) of forms in ${\mathbb {K}}[x,z]$ which generate ${\mathcal {I}}_F$. We give a significantly shorter proof that accumulating all forms for F in our construction yields a minimal grlex Groebner basis ${\mathcal {F}}$ for ${\mathcal {I}}_F$ (without using the theory of S polynomials or distinguishing three types of inverse forms) and we simplify the reduction of ${\mathcal {F}}$. The associated Groebner basis algorithm terminates by construction and is quadratic. Finally we show that $f_1,f_2$ is a maximal ${\mathbb {K}}[x,z]$ regular sequence for ${\mathcal {I}}_F$ , so that ${\mathcal {I}}_F$ is a complete intersection.