I have an applied problem that deals with real multivariate polynomials systems (real coefficients with real roots).
If I understand this problem correctly, the (real) root numbers will remain unchanged within each topologically connected component (please correct me if I'm using a weird terminology here).
$\color{red}{\star}$ Is the real solutions to a polynomial system continuous with respect to the coefficients (in a correspondence continuity sense if variety not zero-dimensional, meaning not finite here), at least within each connected component?
In the finite solutions case, if the solutions are continuous(complex or not), and root numbers are fixed within the component. Then if I just focus on the real ones, it seems like we automatically have continuity for the real solutions. However, this system's solutions are not required to be zero-dimensional. So far, all the books I found only talks about the zero-dimensional case. Would the continuity still hold? Any comments or references are welcomed!