Roots of random polynomials: explaining "holes" in zero plots

Develop general results that characterize and explain the appearance of sparse regions ("holes") in the complex-plane plots of zeros of random polynomials, including conditions on coefficient distributions and degrees that guarantee such phenomena or their absence.

Background

In the course of comparing knot polynomial roots to those of random polynomials, the authors note visual patterns such as apparent holes in zero distributions. While specific cases have been analyzed for certain coefficient sets, a general theory describing when and why such holes arise is not currently available in the literature cited by the authors.

This question, though outside the core quantum invariant comparisons, is raised as a broader mathematical gap encountered during the analysis of root plots.