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.