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Erratum to: manuscripta math. 146, 559–574 (2015) https://doi.org/10.1007/s00229-014-0703-9
This is a corrigendum for the proof of [3, Corrollary 6.1]. We follow the notations in [3]. In the proof of [3, Corrollary 6.1], we claimed that the set \(\mathscr {G}_1\), which is the set of homogeneous polynomials of ST type, is a Zariski closed subset of \(\mathbb {P}(S_{n,d})\). This is incorrect, since, for instance, the polynomial \(f=x^2y-y^2z=y(xy-yz)\in \mathbb {P}(S_{2,3})\) lies in \(\overline{\mathscr {G}}_1\setminus \mathscr {G}_1\), as is proved in [1, Example 1.4]. In fact, in our proof of [3, Corrollary 6.1], an error arises when we obtain the equality (6.3) by letting \(i\rightarrow \infty \) which says
however, it may happen that
thus the right-hand side does not make sense.
On the other hand, the result stated in [3, Corollary 6.1] is correct. Indeed, using [1, Theorem 4.5] and the remark following [2, Theorem 3.2], we obtain \(\overline{\mathscr {G}}_1\setminus \mathscr {G}_1\subset \mathscr {G}_2\), the notation in [3] being used. Then [3, Corollary 6.1] immediately follows from this.
The author would like to thank Professor Hualin Huang and Professor Yu Ye. They asked him to give a talk about the Zariski closedness of \(\mathscr {G}_1\) and after the talk, they told him they had a counterexample at hand. The author eventually found out the error in his original proof.
References
Buczyńska, W., Buczyński, J., Kleppe, J., Teitler, Z.: Apolarity and direct sum decomposability of polynomials. Mich. Math. J. 64, 675–719 (2015)
Fedorchuk, M.: Direct sum decomposability of polynomials and factorization of associated forms. Proc. Lond. Math. Soc. (3) 120(3), 305–327 (2020)
Wang, Z.: On homogeneous polynomials determined by their Jacobian ideal. Manuscr. Math. 146(3–4), 559–574 (2015)
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Wang, Z. Erratum to: On homogeneous polynomials determined by their Jacobian ideal. manuscripta math. 171, 369–370 (2023). https://doi.org/10.1007/s00229-022-01447-2
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DOI: https://doi.org/10.1007/s00229-022-01447-2