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Let

$f_1(x_1,\cdots,x_n)=0$,

$\vdots$

$f_n(x_1,\cdots,x_n)=0,$

be a nonlinear equation. Is there a condition on $f_1,\cdots,f_n$ under which this equation has a solution?

Thanks for your help.

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  • $\begingroup$ For generic $f$? Of course not. $\endgroup$ – Vim Jun 9 '16 at 8:52
  • $\begingroup$ Are you assuming that $f_1,\dots,f_n$ are polynomials? If so, then there is Hilbert Nullstellensatz: there is a solution if and only if $1\notin\left\langle f_1,\dots,f_n\right\rangle$. $\endgroup$ – Guy Jun 9 '16 at 9:03

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