I'm interested in the following question. Please forgive me if my question is lacking in precision. I'm not a mathematician, and need some help getting started:
If I have a smooth, simple curve $\gamma:[0,1]\to \mathbb{C}$, $\gamma (0)=\gamma (1)$, and I apply a polynomial $p: \mathbb{C}\to \mathbb{C}$ to it, must the image $p \circ \gamma$ bound an immersed disc in $\mathbb{C}$?
A thought: If the above is not true in general, can I perturb $\gamma$ slightly to $\gamma + \epsilon f$ (where $f$ is defined on $[0,1]$) such that $p\circ (\gamma + \epsilon f)$ does bound an immersed disc in $\mathbb{C}$? If I perturbe $\gamma$, I might give it a singularity (a loop, for example). I'm not sure what types of singularities survive under polynomials though. Is there a classification for such stable singularities? I can visualize a loop being stable, but is that it?
Take the polynomial $z^2$ and apply it to any circle centered at zero. The image loop will not bound an immersed disk even after perturbation. This means that the resulting map of the circle will not extend to an immersion of the disk.