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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.

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    I am aware that the zeros of a system are not often a discrete set. I wish to know what conditions does F need to satisfy to ensure this.2012-03-29

3 Answers 3

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What you are asking about is whether the ideal generated by the polynomials of the system is zero dimensional. This is a difficult condition to test, which requires the computation of a Gröbner basis for the ideal. Certainly the fact of having as many polynomials as indeterminates, as your question suggest, does not suffice, even if the set of polynomial equations is independent (none of the is implied by the others).

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Example: $F(z_1,z_2)=(z_1-z_2, z_1^2-z_2^2)$; many roots, not isolated.

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    @Suresh: Then have a look at Paxinum's answer above. I found the links provided therein quite enlightening. It seems, however, that those are some strong algebraic conditions. You may be dealing with a system that is easier to analyze (i.e. maybe you don't need the heavy machinery of general theory). Maybe you should tell us more about the specific systems you're thinking about?2012-03-29
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I believe you have a specific system in mind. What you seek is to show that your system of polynomials is a Complete intersection, http://en.wikipedia.org/wiki/Complete_intersection

There are various computer methods for doing this. I am not an expert in this field, but there are sort of generalized Betti numbers, http://en.wikipedia.org/wiki/Betti_number that you may compute, and from those, see how your variety looks like.