Appears in collection : 2026 - T1 - WS3 - Integrating Research and Illustration in Number Theory
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.
By Sébastien Labbé
By Claire David
