Let suppose we have some polynomial system with a lot of real parameters in coefficients. For some parameters the solution space has finite number of points, for some it has one-dimensional solution curve. I am interested particularly in the following question.
We get bounds $N$ for solutions using Bezout theorem. The number $N$ is not large. Then, once we have chosen parameters (when working with the system on practice) we know that it has $N+1$ real solutions. Does it mean that now we have real interval of solutions? According to Bezout theorem, since the solutions are more than bound, it violates the assumption that solution space is finite-dimensional. But it could be complex curve of solutions and we are interested only in real solutions..
It would be very nice if someone suggests how this problem could be attacked.
P.S: to be more precise, the system is following: $$ \begin{cases} w^4 v^4 + F_1(w,v) = 0,\\ w^4 v^4 + F_2(w,v) = 0 \end{cases}.$$
