Restricted polynomial roots, reciprocal power series, and finite capture for collinear affine IFS

Abstract: 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.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper )

---

One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

For questions or subscribing to the mailing list, contact the organisers at numeration@irif.fr

---