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I have a parametric variety produced by the intersection of two multinomials $V_1=0$ and $V_2=\lambda$ where $\lambda$ is a real-valued parameter. I know that for a range of $\lambda \in (\lambda_{min}, \lambda_{max})$ the intersection is not empty. Indeed the intersection can be a curve or perhaps a higher-dimensional object. What I am interested in is the number of connected components of the resulting variety in terms of $\lambda$. More precisely I would like to know if there is a way to prove that for the given range of $\lambda$ the number of connceted components does not change.

Thanks

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  • $\begingroup$ Over what field are you working? Are you working over the field of real numbers? $\endgroup$
    – Jason Starr
    Nov 16, 2015 at 13:04
  • $\begingroup$ It seems from your description that your variety is a level set of some function. So a proper tool should be Morse theory (or some variant of it according to the particular form of your problem). $\endgroup$
    – Pietro Majer
    Nov 19, 2015 at 7:13
  • $\begingroup$ @Pietro: I know a little about Morse theory. My main difficulty is that even with MT, at least as far as I know, the problem is transformed to some condition on the gradient of a parametric polynomial. I do not know how one can easily analyze or check those conditions for the resulting parametric variety. $\endgroup$
    – Saeid Haghighatshoar
    Nov 20, 2015 at 9:28

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