I will discuss how roots of polynomials with coefficients in a finite symmetric integer digit set can be studied through a family of affine iterated function systems whose first-level pieces are centered on a line, evenly spaced, and symmetric with respect to the origin. After passing to reciprocal power series, the closure of the relevant root sets outside the unit disk can be interpreted as a connectedness locus: zeros of such power series describe overlaps in the attractor. This viewpoint is related to the theory of planar self-affine tiles with collinear digit sets, as studied by Akiyama, Loridant, Thuswaldner, and the broader self-affine-tile and number-system literature.

The main part of the talk will explain a finite-capture procedure for the non-real part of the locus, based on successive geometric enclosures. In a precise parameter range, this turns an infinite analytic problem into a finite geometric one. I will emphasize examples and pictures, with only minimal formulas.

This is based on joint work with David Juher, building on earlier joint work with David Juher and Joan Saldaña.