David Savitt and some REU students write about "harmonic curves" and "meanders".  I have not read the estimates in details and I heard they can be optimized.

* [Harmonic algebraic curves and noncrossing partitions](http://front.math.ucdavis.edu/0511.5248)

* [Polynomials, meanders, and paths in the lattice of noncrossing partitions](http://front.math.ucdavis.edu/0606.5169)

The idea is to check $\mathrm{Re}[p(x)]=0$ and $\mathrm{Im}[p(x)]=0$ and to shoe they intersect at exactly $n = \mathrm{deg} \, p$ points.  Then we can have a topological proof since can split the plane into $\mathbb{C} \backslash \mathbb{D} \cup \mathbb{D}$ where on the outside we have an alternating series of lines extending out to infinity and inside we have some kind of non-crossing matching of the lines.

![](https://i.sstatic.net/3iFYN.png)

They come across an objection similar to what you have, **is that topological picture correct**?  Can't the curves $I$ or $R$ be self-intersecting or worse?  

Back then, there was no systematic study of shape such as topology.  Savitt merely proves these curves are *non-singular* without explicitly ruling out these phenomena.

Then Savitt studies $\mathrm{Re}[e^{-i\theta}p(x)] = 0$ which contains $R$ and $I$ as special cases (set $\theta = 0, 90^\circ$)