Let $R = \mathbb{C}\{x_1, \dots,x_n\}$ be the ring of germs of analytic functions and let $f \in R$ be a homogeneus polynomial of degree $p$ such that $\sqrt{Jac(f)}=\langle x_1, \dots, x_{n-1}\rangle \subset R$. If we know that $Jac(f) \subset R$ is a primary ideal, can we conclude that $\sqrt{\langle \partial_1 (f), \dots, \partial_{n-1}(f) \rangle}= \langle x_1, \dots, x_{n-1}\rangle$ as well?
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$\begingroup$ Just to clarify: by $\text{Jac}(f)$ do you mean the ideal generated by $f$ and its first partial derivatives? $\endgroup$– Jason StarrCommented May 17, 2022 at 10:49
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$\begingroup$ Yes, and since $f$ is homogeneous, $Jac(f)$ is simply generated by the partial derivatives of $f$. $\endgroup$– Serge the ToasterCommented May 17, 2022 at 10:54
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$\begingroup$ Aha, I missed the hypothesis that $f$ is homogeneous. $\endgroup$– Jason StarrCommented May 17, 2022 at 11:13
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$\begingroup$ Although I am not sure that this condition is really important for the actual question, I left it since it might be easier to solve. $\endgroup$– Serge the ToasterCommented May 17, 2022 at 11:31
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