How does one show that the polynomial system $F(x)=0,$ where $F:\mathbb{C}^n \rightarrow \mathbb{C}^n,$ has only isolated roots?
As an example, let $\mathcal{E}(\mathbf{x})\equiv(e_{1}(\mathbf{x}),e_{2}(\mathbf{x}),e_{3}(\mathbf{x}),e_{4}(\mathbf{x})),$ where $\mathbf{x}\equiv(x_{1},x_{2},x_{3},x_{4})$ and
$e_{1}(\mathbf{x}) = x_{1}+x_{3}+5(x_{1}x_{4}+x_{2}x_{3})$
$e_{2}(\mathbf{x}) = x_{1}x_{3};$
$e_{3}(\mathbf{x}) = x_{2}+x_{4}-6(x_{1}x_{4}+x_{2}x_{3});$
$e_{4}(\mathbf{x}) = x_{2}x_{4}$
An easy result to show is that the set $\mathcal{S}=\left\{ \mathbf{x}\in\mathbb{C}^{4}:\mathcal{E}^{\prime}(\mathbf{x})\mbox{ is nonsingular}\right\} $ has measure zero. With $\mathbf{w}\equiv(w_{1},w_{2},w_{3},w_{4})\in\mathcal{S},$ consider the polynomial system given by \begin{equation} H(\mathbf{x})=\mathcal{E}(\mathbf{x})-\mathcal{E}(\mathbf{w}) = 0. \end{equation} For a family of such systems, I wish to know whether there are finite roots. I would like to determine properties that ensure the roots are finite.