Consider the $\mathbb C$-algebra $\mathcal O_2$ of holomorphic function germs on $\mathbb C^2$ at the origin $0$. Is it true that any germ $f\in\mathcal O_2$ with $f(0)=0$ belongs to $(\partial f/\partial x,\partial f/\partial y)$, the ideal generated by its partial derivatives?
2026-03-25 06:51:39.1774421499
Does any function germ belong to its gradient ideal?
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