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04-06-2019 | Original Paper | Issue 6/2019

Applicable Algebra in Engineering, Communication and Computing 6/2019

On the annihilator ideal of an inverse form: addendum

Journal:
Applicable Algebra in Engineering, Communication and Computing > Issue 6/2019
Author:
Graham H. Norton
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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.

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