Semialgebraic interior layers in root closures of bounded-height integer polynomials

Abstract

Finite-degree root plots of bounded-height monic integer polynomials can exhibit gaps around interior points of their root closures. Outside the closed unit disk, these closures coincide with Mandelbrot sets for affine iterated function systems with collinear digits. We construct explicit semialgebraic interior layers, certified when finite inverse orbits enter canonical self-covering parallelograms. Such parallelograms exist exactly in the non-real lens, which contains the non-real locus precisely from twenty maps onward. Within the non-real lens, limits of each capture level are captured within two additional steps, optimally, and the layers recover the locus by closure. The layers expose persistent interior deficits: explicit quadratic parameters evade every finite level; sufficiently close distinct parameters have logarithmically growing capture depths; the underlying lattice obstruction yields degree-dependent root-free radii, explaining such gaps. For non-real parameters of modulus at least the square root of the number of maps, exact enclosure trees have sharp, alphabet-independent branching bounds.

47 pages, 15 figures. Substantially revised and retitled, with additional results. Computational materials: https://doi.org/10.5281/zenodo.23441158 ; interactive explorer: https://complextrees.com/collinear/