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Topic: Parametric Polynomial System... How to single out a specific root?
Replies: 1   Last Post: Apr 8, 2013 3:31 PM

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quasi

Posts: 10,314
Registered: 7/15/05
Re: Parametric Polynomial System... How to single out a specific root?
Posted: Apr 8, 2013 3:31 PM
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golabidoon wrote:
>
>I have a system of n polynomial equations in n variables:
>
>f_k(x_1,x_2,...,x_n;t)=0 for k=1 to n.
>
>Denote X=(x_1,x_2,...x_n)
>
>These equations all depends on a parameter 0<=t<=1. When t=0,
>then the system has only one simple (no multiplicity) finite
>real root X^*, and that root is easy to compute.
>
>When t=1, then the problem may have different roots with
>multiplicity, and the roots are difficult to compute. I am
>hence not interested in all the real roots when t=1, but only
>a particular one specified below.
>
>Due to continuity of roots w.r.t. to polynomial coefficients,
>we know that each (possibly complex) root of the system at
>side t=0 has a corresponding root when t=1. I am only
>interested in the root at side t=1 which corresponds to the
>single finite and simple root of the system when t=0.
>
>Is there a way to characterize this root? I need an analytical
>form, either exact or approximate, as this is a part of my
>analysis to establish some bounds. So, very relevant numerical
>schemes such as homotopy continunation are unfortunately not
>an option here.


A tough ask.

Can you even resolve the case n=1? If so, how? If not,
forget it.

quasi



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