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Jul 2, 2021 at 8:11 vote accept user90189
Jun 29, 2021 at 18:51 comment added user90189 I really do appreciate this answer, I feel relief. So, to check if the variety is smooth, I can try proving that $x_i^N = RQ_d + R_1\partial_1Q_d + \cdots R_n\partial_n Q_d$ for some $N$ and every $i$. This condition is very interesting, I've came across this sort of condition when proving that a Sobolev space is equivalent to other function space with norm $\sum_i\lVert Q_i(D)f\rVert_p$. Sorry if I don't accept it right away, I prefer to wait.
Jun 29, 2021 at 16:25 history answered Will Sawin CC BY-SA 4.0