# On the annihilator ideal of an inverse form: addendum

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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.

## Keywords

Annihilator ideal Berlekamp–Massey algorithm Complete intersection Grlex Groebner basis Inverse form Regular sequence## Notes

## References

- 1.Althaler, J., Dür, A.: Finite linear recurring sequences and homogeneous ideals. Appl. Algebra Eng. Commun. Comput.
**7**, 377–390 (1996)MathSciNetzbMATHGoogle Scholar - 2.Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms. UTM, 3rd edn. Springer, Berlin (2007)CrossRefGoogle Scholar
- 3.Massey, J.L.: Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory
**15**, 122–127 (1969)MathSciNetCrossRefGoogle Scholar - 4.Northcott, D.G.: Injective envelopes and inverse polynomials. J. Lond. Math. Soc.
**8**, 290–296 (1974)MathSciNetCrossRefGoogle Scholar - 5.Norton, G.H.: On the annihilator ideal of an inverse form. J. Appl. Algebra Eng. Commun. Comput.
**28**, 31–78 (2017)MathSciNetCrossRefGoogle Scholar - 6.Norton, G.H.: On the annihilator ideal of an inverse form. ArXiv:1710.07731 (2017)
- 7.Sharp, R.Y.: Steps in Commutative Algebra, 2nd edn. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar