Orateur
Description
Fix n and consider the roots of polynomials whose coefficients are integers with absolute value at most n−1. Taken over all degrees, these roots form a countable set of algebraic numbers, but their closure has a striking fractal geometry. I will explain how, after a reciprocal-power-series reformulation, the problem becomes a connectedness question for a family of self-similar sets. The key new idea is a finite-capture depth filtration built from a canonical trap-and-enclosure construction in a natural two-disk lens region of parameter space. The level Θₖ(n) consists of parameters that can be certified by following a single marked point for at most k inverse steps. The main result shows that these layers fit together with uniform regularity: every limit of depth-k parameters already lies in depth k+2. In the lens, closing up the finite-capture locus recovers the entire non-real closure of the set of roots, and for n≥20 this yields the full non-real picture. The talk will emphasize the geometry and illustrations behind this finite organization of a fractal closure of algebraic numbers.
The corresponding preprint is available at arXiv:2603.07397
https://arxiv.org/abs/2603.07397