I am dealing with the following situation:
I have a real analytic system of equations, $F(x,t)=0$, with $F: [0,1]^n \times [0, \infty) \rightarrow \mathbb{R}^n $.
I know that only a single solution, $x_0$ exists for $t=0$, and that this solution $(x_0,0)$ is not a critical point of $F$. This means that also for $t$ sufficiently small, a unique solution exists.
No other solutions exist on the boundary of the domain.
I wish to show that for any given $t>0$, only isolated solutions exist.
(If I'm not mistaken, this also implies that the solution space to $F(x,t)=0$ consists of (possibly intersecting) curves and isolated points only. And, that the vector field spanned by $F$ for any fixed $t$ has isolated equilibria only - please correct if necessary).
I realize that one way forward is considering the Jacobian of $F$ and the points where it does not have full rank; but so far that has not proven very fruitful, as this just shifts the problem to showing that the system of its subdeterminants only has isolated solutions - in the given case, that does not seem to be easier.
If I'm not mistaken, the property I'm looking should be generic due to Sard's theorem - unfortunately that is not enough for my purposes.
Are there any other approaches or results that could be promising?
I feel that the Identity theorem for real analytic functions might be of help, but I'm having trouble how make use of it here.
Edit: This question on zero sets of real analytic functions is likely related as well. Still trying to understand if and how the discussion and sources cited there might help for my problem.
Happy to fill in more details on $F$ if helpful.
